keyword:hard - cp's OEIS frontend

A000043 Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281

Offset: 1

N. J. A. Sloane

Equivalently, integers k such that 2^k - 1 is prime.
It is believed (but unproved) that this sequence is infinite. The data suggest that the number of terms up to exponent N is roughly K log N for some constant K.
Length of prime repunits in base 2.
The associated perfect number N=2^(p-1)*M(p) (=A019279*A000668=A000396), has 2p (=A061645) divisors with harmonic mean p (and geometric mean sqrt(N)). - Lekraj Beedassy, Aug 21 2004
In one of his first publications Euler found the numbers up to 31 but erroneously included 41 and 47.
Equals number of bits in binary expansion of n-th Mersenne prime (A117293). - Artur Jasinski, Feb 09 2007
Number of divisors of n-th even perfect number, divided by 2. Number of divisors of n-th even perfect number that are powers of 2. Number of divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n). - Omar E. Pol, Feb 24 2008
Number of divisors of n-th even superperfect number A061652(n). Numbers of divisors of n-th superperfect number A019279(n), assuming there are no odd superperfect numbers. - Omar E. Pol, Mar 01 2008
Differences between exponents when the even perfect numbers are represented as differences of powers of 2, for example: The 5th even perfect number is 33550336 = 2^25 - 2^12 then a(5)=25-12=13 (see A135655, A133033, A090748). - Omar E. Pol, Mar 01 2008
Number of 1's in binary expansion of n-th even perfect number (see A135650). Number of 1's in binary expansion of divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n) (see A135652, A135653, A135654, A135655). - Omar E. Pol, May 04 2008
Indices of the numbers A006516 that are also even perfect numbers. - Omar E. Pol, Aug 30 2008
Indices of Mersenne numbers A000225 that are also Mersenne primes A000668. - Omar E. Pol, Aug 31 2008
The (prime) number p appears in this sequence if and only if there is no prime q<2^p-1 such that the order of 2 modulo q equals p; a special case is that if p=4k+3 is prime and also q=2p+1 is prime then the order of 2 modulo q is p so p is not a term of this sequence. - Joerg Arndt, Jan 16 2011
Primes p such that sigma(2^p) - sigma(2^p-1) = 2^p-1. - Jaroslav Krizek, Aug 02 2013
Integers k such that every degree k irreducible polynomial over GF(2) is also primitive, i.e., has order 2^k-1.  Equivalently, the integers k such that A001037(k) = A011260(k). - Geoffrey Critzer, Dec 08 2019
Conjecture: for k > 1, 2^k-1 is (a Mersenne) prime or k = 2^(2^m)+1 (is a Fermat number) if and only if (k-1)^(2^k-2) == 1 (mod (2^k-1)k^2). - Thomas Ordowski, Oct 05 2023
Conjecture: for p prime, 2^p-1 is (a Mersenne) prime or p = 2^(2^m)+1 (is a Fermat number) if and only if (p-1)^(2^p-2) == 1 (mod 2^p-1). - David Barina, Nov 25 2024
Already as of Dec. 2020, all exponents up to 10^8 had been verified, implying that 74207281, 77232917 and 82589933 are indeed the next three terms. As of today, all exponents up to 130439863 have been tested at least once, see the GIMPS Milestones Report. - M. F. Hasler, Apr 11 2025
On June 23. 2025 all exponents up to 74340751 have been verified, confirming that 74207281 is the exponent of the 49th Mersenne Prime. - Rodolfo Ruiz-Huidobro, Jun 23 2025

			Corresponding to the initial terms 2, 3, 5, 7, 13, 17, 19, 31 ... we get the Mersenne primes 2^2 - 1 = 3, 2^3 - 1 = 7, 2^5 - 1 = 31, 127, 8191, 131071, 524287, 2147483647, ... (see A000668).

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.
J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 79.
R. K. Guy, Unsolved Problems in Number Theory, Section A3.
F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 57.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 19.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 132-134.
B. Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun, 1971), Abstract 684-A15, p. 608.

P. T. Bateman, J. L. Selfridge, and S. S. Wagstaff, Jr., The new Mersenne conjecture, Amer. Math. Monthly 96 (1989), no. 2, 125--128. MR0992073 (90c:11009).
J. Bernheiden, Mersenne Numbers (Text in German)
Andrew R. Booker, The Nth Prime Page
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
P. G. Brown, A Note on Ramanujan's (FALSE) Conjectures Regarding 'Mersenne Primes'
C. K. Caldwell, Mersenne Primes
C. K. Caldwell, Recent Mersenne primes
Zuling Chang, Martianus Frederic Ezerman, Adamas Aqsa, Fahreza, San Ling, Janusz Szmidt, and Huaxiong Wang, Binary de Bruijn Sequences via Zech's Logarithms, 2018.
Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
Leonhard Euler, Observations on a theorem of Fermat and others on looking at prime numbers, arXiv:math/0501118 [math.HO], 2005-2008.
Leonhard Euler, Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus
G. Everest et al., Primes generated by recurrence sequences, arXiv:math/0412079 [math.NT], 2006.
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Luis H. Gallardo and Olivier Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.
Donald B. Gillies, Three new Mersenne primes and a statistical theory Mathematics of Computation 18.85 (1964): 93-97.
GIMPS (Great Internet Mersenne Prime Search), Distributed Computing Projects
GIMPS, Milestones Report
GIMPS, GIMPS Project discovers largest known prime number 2^77232917-1
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
H. Lifchitz, Mersenne and Fermat primes field
A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see p. 143.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
Romeo Meštrović, Goldbach-type conjectures arising from some arithmetic progressions, University of Montenegro, 2018.
Romeo Meštrović, Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes, arXiv:1901.07882 [math.NT], 2019.
G. P. Michon, Perfect Numbers, Mersenne Primes
Albert A. Mullin, Letter to the editor, about "The new Mersenne conjecture" [Amer. Math. Monthly 96 (1989), no. 2, 125-128; MR0992073 (90c:11009)] by P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, Jr., Amer. Math. Monthly 96 (1989), no. 6, 511. MR0999415 (90f:11008).
Curt Noll and Laura Nickel, The 25th and 26th Mersenne primes, Math. Comp. 35 (1980), 1387-1390.
M. Oakes, A new series of Mersenne-like Gaussian primes
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
H. J. Smith, Mersenne Primes
B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68 (1971), 2319-2320.
H. S. Uhler, On All Of Mersenne's Numbers Particularly M_193, PNAS 1948 34 (3) 102-103.
H. S. Uhler, First Proof That The Mersenne Number M_157 Is Composite, PNAS 1944 30(10) 314-316.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Cunningham Number
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Mersenne Prime
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Wagstaff's Conjecture
David Whitehouse, Number takes prime position (2^13466917 - 1 found after 13000 years of computer time)
K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte für Mathematik und Physik, Vol. 3, No. 1 (1892), 265-284.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime
Index entries for "core" sequences

Cf. A000668 (Mersenne primes).
Cf. A028335 (integer lengths of Mersenne primes).
Cf. A000225 (Mersenne numbers).
Cf. A001348 (Mersenne numbers with n prime).
Cf. A016027, A046051, A057429, A057951-A057958, A066408, A117293, A127962, A127963, A127964, A127965, A127961, A000979, A000978, A124400, A124401, A127955, A127956, A127957, A127958, A127936, A134458, A000225, A000396, A090748, A133033, A135655, A006516, A019279, A061652, A133033, A135650, A135652, A135653, A135654, A260073, A050475, A379590.

MersennePrimeExponent[Range[48]] (* Eric W. Weisstein, Jul 17 2017; updated Oct 21 2024 *)

isA000043(n) = isprime(2^n-1) \\ Michael B. Porter, Oct 28 2009

is(n)=my(h=Mod(2,2^n-1)); for(i=1, n-2, h=2*h^2-1); h==0||n==2 \\ Lucas-Lehmer test for exponent e. - Joerg Arndt, Jan 16 2011, and Charles R Greathouse IV, Jun 05 2013
forprime(e=2,5000,if(is(e),print1(e,", "))); /* terms < 5000 */

from sympy import isprime, prime
for n in range(1,100):
    if isprime(2**prime(n)-1):
        print(prime(n), end=', ') # Stefano Spezia, Dec 06 2018

a(n) = log((1/2)*(1+sqrt(1+8*A000396(n))))/log(2). - Artur Jasinski, Sep 23 2008 (under the assumption there are no odd perfect numbers, Joerg Arndt, Feb 23 2014)
a(n) = A000005(A061652(n)). - Omar E. Pol, Aug 26 2009
a(n) = A000120(A000396(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Oct 30 2013

Also in the sequence: p = 74207281. - Charles R Greathouse IV, Jan 19 2016
Also in the sequence: p = 77232917. - Eric W. Weisstein, Jan 03 2018
Also in the sequence: p = 82589933. - Gord Palameta, Dec 21 2018
a(46) = 42643801 and a(47) = 43112609, whose ordinal positions in the sequence are now confirmed, communicated by Eric W. Weisstein, Apr 12 2018
a(48) = 57885161, whose ordinal position in the sequence is now confirmed, communicated by Benjamin Przybocki, Jan 05 2022
Also in the sequence: p = 136279841. - Eric W. Weisstein, Oct 21 2024
As of Jan 31 2025, 48 terms are known, and are shown in the DATA section. Four additional numbers are known to be in the sequence, namely 74207281, 77232917, 82589933, and 136279841, but they may not be the next terms. See the GIMP website for the latest information. - N. J. A. Sloane, Jan 31 2025

A000668 Mersenne primes (primes of the form 2^n - 1).

Original entry on oeis.org

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

N. J. A. Sloane

For a Mersenne number 2^n - 1 to be prime, the exponent n must itself be prime.
See A000043 for the values of n.
Primes that are repunits in base 2.
Define f(k) = 2k+1; begin with k = 2, a(n+1) = least prime of the form f(f(f(...(a(n))))). - Amarnath Murthy, Dec 26 2003
Mersenne primes other than the first are of the form 6n+1. - Lekraj Beedassy, Aug 27 2004. Mersenne primes other than the first are of the form 24n+7; see also A124477. - Artur Jasinski, Nov 25 2007
A034876(a(n)) = 0 and A034876(a(n)+1) = 1. - Jonathan Sondow, Dec 19 2004
Mersenne primes are solutions to sigma(n+1)-sigma(n) = n as perfect numbers (A000396(n)) are solutions to sigma(n) = 2n. In fact, appears to give all n such that sigma(n+1)-sigma(n) = n. - Benoit Cloitre, Aug 27 2002
If n is in the sequence then sigma(sigma(n)) = 2n+1. Is it true that this sequence gives all numbers n such that sigma(sigma(n)) = 2n+1? - Farideh Firoozbakht, Aug 19 2005
It is easily proved that if n is a Mersenne prime then sigma(sigma(n)) - sigma(n) = n. Is it true that Mersenne primes are all the solutions of the equation sigma(sigma(x)) - sigma(x) = x? - Farideh Firoozbakht, Feb 12 2008
Sum of divisors of n-th even superperfect number A061652(n). Sum of divisors of n-th superperfect number A019279(n), if there are no odd superperfect numbers. - Omar E. Pol, Mar 11 2008
Indices of both triangular numbers and generalized hexagonal numbers (A000217) that are also even perfect numbers. - Omar E. Pol, May 10 2008, Sep 22 2013
Number of positive integers (1, 2, 3, ...) whose sum is the n-th perfect number A000396(n). - Omar E. Pol, May 10 2008
Vertex number where the n-th perfect number A000396(n) is located in the square spiral whose vertices are the positive triangular numbers A000217. - Omar E. Pol, May 10 2008
Mersenne numbers A000225 whose indices are the prime numbers A000043. - Omar E. Pol, Aug 31 2008
The digital roots are 1 if p == 1 (mod 6) and 4 if p == 5 (mod 6). [T. Koshy, Math Gaz. 89 (2005) p. 465]
Primes p such that for all primes q < p, p XOR q = p - q. - Brad Clardy, Oct 26 2011
All these primes, except 3, are Brazilian primes, so they are also in A085104 and A023195. - Bernard Schott, Dec 26 2012
All prime numbers p can be classified by k = (p mod 12) into four classes: k=1, 5, 7, 11. The Mersennne prime numbers 2^p-1, p > 2 are in the class k=7 with p=12*(n-1)+7, n=1,2,.... As all 2^p (p odd) are in class k=8 it follows that all 2^p-1, p > 2 are in class k=7. - Freimut Marschner, Jul 27 2013
From "The Guinness Book of Primes": "During the reign of Queen Elizabeth I, the largest known prime number was the number of grains of rice on the chessboard up to and including the nineteenth square: 524,287 [= 2^19 - 1]. By the time Lord Nelson was fighting the Battle of Trafalgar, the record for the largest prime had gone up to the thirty-first square of the chessboard: 2,147,483,647 [= 2^31 - 1]. This ten-digits number was proved to be prime in 1772 by the Swiss mathematician Leonard Euler, and it held the record until 1867." [du Sautoy] - Robert G. Wilson v, Nov 26 2013
If n is in the sequence then A024816(n) = antisigma(n) = antisigma(n+1) - 1. Is it true that this sequence gives all numbers n such that antisigma(n) = antisigma(n+1) - 1? Are there composite numbers with this property? - Jaroslav Krizek, Jan 24 2014
If n is in the sequence then phi(n) + sigma(sigma(n)) = 3n. Is it true that Mersenne primes are all the solutions of the equation phi(x) + sigma(sigma(x)) = 3x? - Farideh Firoozbakht, Sep 03 2014
a(5) = A229381(2) = 8191 is the "Simpsons' Mersenne prime". - Jonathan Sondow, Jan 02 2015
Equivalently, prime powers of the form 2^n - 1, see Theorem 2 in Lemos & Cambraia Junior. - Charles R Greathouse IV, Jul 07 2016
Primes whose sum of divisors is a power of 2. Primes p such that p + 1 is a power of 2. Primes in A046528. - Omar E. Pol, Jul 09 2016
From Jaroslav Krizek, Jan 19 2017: (Start)
Primes p such that sigma(p+1) = 2p+1.
Primes p such that A051027(p) = sigma(sigma(p)) = 2^k-1 for some k > 1.
Primes p of the form sigma(2^prime(n)-1)-1 for some n. Corresponding values of numbers n are in A016027.
Primes p of the form sigma(2^(n-1)) for some n > 1. Corresponding values of numbers n are in A000043 (Mersenne exponents).
Primes of the form sigma(2^(n+1)) for some n > 1. Corresponding values of numbers n are in A153798 (Mersenne exponents-2).
Primes p of the form sigma(n) where n is even; subsequence of A023195. Primes p of the form sigma(n) for some n. Conjecture: 31 is the only prime p such that p = sigma(x) = sigma(y) for distinct numbers x and y; 31 = sigma(16) = sigma(25).
Conjecture: numbers n such that n = sigma(sigma(n+1)-n-1)-1, i.e., A072868(n)-1.
Conjecture: primes of the form sigma(4*(n-1)) for some n. Corresponding values of numbers n are in A281312. (End)
[Conjecture] For n > 2, the Mersenne number M(n) = 2^n - 1 is a prime if and only if 3^M(n-1) == -1 (mod M(n)). - Thomas Ordowski, Aug 12 2018 [This needs proof! - Joerg Arndt, Mar 31 2019]
Named "Mersenne's numbers" by W. W. Rouse Ball (1892, 1912) after Marin Mersenne (1588-1648). - Amiram Eldar, Feb 20 2021
Theorem. Let b = 2^p - 1 (where p is a prime). Then b is a Mersenne prime iff (c = 2^p - 2 is totient or a term of A002202). Otherwise, if c is (nontotient or a term of A005277) then b is composite. Proof. Trivial, since, while b = v^g - 1 where v is even, v > 2, g is an integer, g > 1, b is always composite, and c = v^g - 2 is nontotient (or a term of A005277), and so is for any composite b = 2^g - 1 (in the last case, c = v^g - 2 is also nontotient, or a term of A005277). - Sergey Pavlov, Aug 30 2021 [Disclaimer: This proof has not been checked. - N. J. A. Sloane, Oct 01 2021]

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.
John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman and S. S. Wagstaff, Jr., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 135-136.
Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 76.
Marcus P. F. du Sautoy, The Number Mysteries, A Mathematical Odyssey Through Everyday Life, Palgrave Macmillan, First published in 2010 by the Fourth Estate, an imprint of Harper Collins UK, 2011, p. 46. - Robert G. Wilson v, Nov 26 2013
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Bryant Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun, 1971), Abstract 684-A15, p. 608.

Harry J. Smith, Table of n, a(n) for n = 1..18
Peter Alfeld, The 39th Mersenne prime, 2003.
Yan Bingze, Li Qiong, Mao Haokun, and Chen Nan, An efficient hybrid hash based privacy amplification algorithm for quantum key distribution, arXiv:2105.13678 [quant-ph], 2021.
Andrew R. Booker, The Nth Prime Page
John Rafael M. Antalan, A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number Puzzle, arXiv:1908.06014 [math.HO], 2019-2020.
W. W. Rouse Ball, Mathematical recreations and problems of past and present times, London, Macmillan and Co., 1892, pp. 24-25.
W. W. Rouse Ball, Mersenne's numbers, Messenger of Mathematics, Vol. 21 (1892), pp. 34-40, 121.
W. W. Rouse Ball, Mersenne's numbers, Nature, Vol. 89 (1912), p. 86.
John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman and S. S. Wagstaff, Jr., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.
Douglas Butler, Mersenne Primes.
C. K. Caldwell, Mersenne primes.
C. K. Caldwell, "Top Twenty" page, Mersenne Primes.
Luis H. Gallardo and Olivier Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 12.
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 15 2012
Abílio Lemos and Ady Cambraia Junior, On the number of prime factors of Mersenne numbers, arXiv:1606.08690 [math.NT] (2016).
Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola (2018) Vol. 54, Issue 3.
Math Reference Project, Mersenne and Fermat Primes.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From _N. J. A. Sloane_, Jun 13 2012
Romeo Meštrović, Goldbach-type conjectures arising from some arithmetic progressions, University of Montenegro, 2018.
Romeo Meštrović, Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes, arXiv:1901.07882 [math.NT], 2019.
Landon Curt Noll, Mersenne Prime Digits and Names.
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Primefan, The Mersenne Primes.
Christian Salas, Cantor Primes as Prime-Valued Cyclotomic Polynomials, arXiv preprint arXiv:1203.3969 [math.NT], 2012.
Harry J. Smith, Mersenne Primes, 1981-2010.
Gordon Spence, 36th Mersenne Prime Found, 1999.
Susan Stepney, Mersenne Prime.
Thesaurus.maths.org, Mersenne Prime.
Bryant Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, Vol. 68 (1971), pp. 2319-2320.
Samuel S. Wagstaff, Jr., The Cunningham Project.
Yunlan Wei, Yanpeng Zheng, Zhaolin Jiang and Sugoog Shon, A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers, Mathematics, Vol. 7, No. 10 (2019), 893.
Eric Weisstein's World of Mathematics, Mersenne Prime.
Eric Weisstein's World of Mathematics, Perfect Number.
Wikipedia, Mersenne prime.
Marek Wolf, Computer experiments with Mersenne primes, arXiv preprint arXiv:1112.2412 [math.NT], 2011.
Chai Wah Wu, Can machine learning identify interesting mathematics? An exploration using empirically observed laws, arXiv:1805.07431 [cs.LG], 2018.

Cf. A000225 (Mersenne numbers).
Cf. A000043 (Mersenne exponents).
Cf. A001348 (Mersenne numbers with n prime).
Cf. A000040, A000203, A000217, A000396, A003056, A007947, A016027, A019279, A023195, A023758, A028335 (lengths), A034876, A046051, A057951-A057958, A059305, A061652, A083420, A085104, A124477, A135659, A173898.

A000668:=Filtered(List(Filtered([1..600], IsPrime),i->2^i-1),IsPrime); # Muniru A Asiru, Oct 01 2017

A000668 := proc(n) local i;
i := 2^(ithprime(n))-1:
if (isprime(i)) then
   return i
fi: end:
seq(A000668(n), n=1..31); # Jani Melik, Feb 09 2011
# Alternate:
seq(numtheory:-mersenne([i]),i=1..26); # Robert Israel, Jul 13 2014

2^Array[MersennePrimeExponent, 18] - 1 (* Jean-François Alcover, Feb 17 2018, Mersenne primes with less than 1000 digits *)
2^MersennePrimeExponent[Range[18]] - 1 (* Eric W. Weisstein, Sep 04 2021 *)

forprime(p=2,1e5,if(ispseudoprime(2^p-1),print1(2^p-1", "))) \\ Charles R Greathouse IV, Jul 15 2011

LL(e) = my(n, h); n = 2^e-1; h = Mod(2, n); for (k=1, e-2, h=2*h*h-1); return(0==h) \\ after Joerg Arndt in A000043
forprime(p=1, , if(LL(p), print1(p, ", "))) \\ Felix Fröhlich, Feb 17 2018

from sympy import isprime, primerange
print([2**n-1 for n in primerange(1, 1001) if isprime(2**n-1)]) # Karl V. Keller, Jr., Jul 16 2020

a(n) = sigma(A061652(n)) = A000203(A061652(n)). - Omar E. Pol, Apr 15 2008
a(n) = sigma(A019279(n)) = A000203(A019279(n)), provided that there are no odd superperfect numbers. - Omar E. Pol, May 10 2008
a(n) = A000225(A000043(n)). - Omar E. Pol, Aug 31 2008
a(n) = 2^A000043(n) - 1 = 2^(A000005(A061652(n))) - 1. - Omar E. Pol, Oct 27 2011
a(n) = A000040(A059305(n)) = A001348(A016027(n)). - Omar E. Pol, Jun 29 2012
a(n) = A007947(A000396(n))/2, provided that there are no odd perfect numbers. - Omar E. Pol, Feb 01 2013
a(n) = 4*A134709(n) + 3. - Ivan N. Ianakiev, Sep 07 2013
a(n) = A003056(A000396(n)), provided that there are no odd perfect numbers. - Omar E. Pol, Dec 19 2016
Sum_{n>=1} 1/a(n) = A173898. - Amiram Eldar, Feb 20 2021

A056654 Numbers k such that 10*R_k + 3 is prime, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 23, 83, 220, 1313, 2951, 20015, 51053

Offset: 1

Robert G. Wilson v, Aug 09 2000

Also numbers k such that (10^(k+1)+17)/9 is prime.

a(14) > 10^5. - Robert Price, Nov 01 2014

			8 is a term because 111111113 is a prime.

Makoto Kamada, Prime numbers of the form 11...113.
Index entries for primes involving repunits

Cf. A093011 (corresponding primes), A097683.

Do[ If[ PrimeQ[ 10*(10^n - 1)/9 + 3 ], Print[ n ] ], {n, 0, 1350} ]

is(n)=ispseudoprime(10^n\9*10+3) \\ Charles R Greathouse IV, Nov 10 2021

a(n) = A097683(n+1) - 1. - Robert Price, Nov 01 2014

a(11) (only a probable prime) from Rick L. Shepherd, Mar 14 2004

a(12)-a(13) derived from A097683 by Robert Price, Nov 01 2014

A019434 Fermat primes: primes of the form 2^(2^k) + 1, for some k >= 0.

Original entry on oeis.org

3, 5, 17, 257, 65537

Offset: 1

N. J. A. Sloane, David W. Wilson

It is conjectured that there are only 5 terms. Currently it has been shown that 2^(2^k) + 1 is composite for 5 <= k <= 32 (see Eric Weisstein's Fermat Primes link). - Dmitry Kamenetsky, Sep 28 2008
No Fermat prime is a Brazilian number. So Fermat primes belong to A220627. For proof see Proposition 3 page 36 in "Les nombres brésiliens" in Links. - Bernard Schott, Dec 29 2012
This sequence and A001220 are disjoint (see "Other theorems about Fermat numbers" in Wikipedia link). - Felix Fröhlich, Sep 07 2014
Numbers n > 1 such that n * 2^(n-2) divides (n-1)! + 2^(n-1). - Thomas Ordowski, Jan 15 2015
From Jaroslav Krizek, Mar 17 2016: (Start)
Primes p such that phi(p) = 2*phi(p-1); primes from A171271.
Primes p such that sigma(p-1) = 2p - 3.
Primes p such that sigma(p-1) = 2*sigma(p) - 5.
For n > 1, a(n) = primes p such that p = 4 * phi((p-1) / 2) + 1.
Subsequence of A256444 and A256439.
Conjectures:
1) primes p such that phi(p) = 2*phi(p-2).
2) primes p such that phi(p) = 2*phi(p-1) = 2*phi(p-2).
3) primes p such that p = sigma(phi(p-2)) + 2.
4) primes p such that phi(p-1) + 1 divides p + 1.
5) numbers n such that sigma(n-1) = 2*sigma(n) - 5. (End)
Odd primes p such that ratio of the form (the number of nonnegative m < p such that m^q == m (mod p))/(the number of nonnegative m < p such that -m^q == m (mod p)) is a divisor of p for all nonnegative q. - Juri-Stepan Gerasimov, Oct 13 2020
Numbers n such that tau(n)*(number of distinct ratio (the number of nonnegative m < n such that m^q == m (mod n))/(the number of nonnegative m < n such that -m^q == m (mod n))) for nonnegative q is equal to 4. - Juri-Stepan Gerasimov, Oct 22 2020
The numbers of primitive roots for the five known terms are 1, 2, 8, 128, 32768. - Gary W. Adamson, Jan 13 2022
Prime numbers such that every residue is either a primitive root or a quadratic residue. - Keith Backman, Jul 11 2022
If there are only 5 Fermat primes, then there are only 31 odd order groups which have a 2-group automorphism group. See the Miles Englezou link for a proof. - Miles Englezou, Mar 10 2025

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 137-141, 197.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see Table 1, p. 458.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 78-79.
Richard K. Guy, Unsolved Problems in Number Theory, A3.
Hardy and Wright, An Introduction to the Theory of Numbers, bottom of page 18 in the sixth edition, gives an heuristic argument that this sequence is finite.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 7, 70.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.

Anas AbuDaqa, Amjad Abu-Hassan, and Muhammad Imam, Taxonomy and Practical Evaluation of Primality Testing Algorithms, arXiv:2006.08444 [cs.CR], 2020.
Cyril Banderier, Pepin's Criterion For Fermat Numbers (in French)
Kent D. Boklan and John H. Conway, Expect at most one billionth of a new Fermat Prime!, arXiv:1605.01371 [math.NT], 2016.
P. Bruillard, S.-H. Ng, E. Rowell, and Z. Wang, On modular categories, arXiv:1310.7050 [math.QA], 2013-2015.
C. K. Caldwell, The Prime Glossary, Fermat number
Miles Englezou, Proof that 5 Fermat primes implies 31 odd order groups with 2-group automorphism groups
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
Romeo Meštrović, Goldbach-type conjectures arising from some arithmetic progressions, University of Montenegro, 2018.
Romeo Meštrović, Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes, arXiv:1901.07882 [math.NT], 2019.
Salah Eddine Rihane, Chèfiath Awero Adegbindin, and Alain Togbé, Fermat Padovan And Perrin Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.2.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 65.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38. Local copy, included here with permission from the editors of Quadrature.
Yaryna Stelmakh, Homeomorphisms of the space of non-zero integers with the Kirch topology, arXiv:2101.04676 [math.GN], 2020.
Eric Weisstein's World of Mathematics, Fermat Number
Eric Weisstein's World of Mathematics, Fermat Prime
Eric Weisstein's World of Mathematics, Pepin's Test
Wikipedia, Fermat number
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A000215, A001146, A159611, A220627, A220570.
Subsequence of A147545 and of A334101. Cf. also A333788, A334092.
Cf. A045544.

[2^(2^n)+1 : n in [0..4] | IsPrime(2^(2^n)+1)]; // Arkadiusz Wesolowski, Jun 09 2011

Select[Table[2^(2^n) + 1, {n, 0, 4}], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

for(i=0,10, isprime(2^2^i+1) && print1(2^2^i+1,", ")) \\ M. F. Hasler, Nov 21 2009

[2^(2^n)+1 for n in (0..4) if is_prime(2^(2^n)+1)] # G. C. Greubel, Mar 07 2019

a(n+1) = A180024(A049084(a(n))). - Reinhard Zumkeller, Aug 08 2010

a(n) = 1 + A001146(n-1), if 1 <= n <= 5. - Omar E. Pol, Jun 08 2018

A007097 Primeth recurrence: a(n+1) = a(n)-th prime.

Original entry on oeis.org

1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, 88362852307, 2428095424619, 75063692618249, 2586559730396077, 98552043847093519, 4123221751654370051, 188272405179937051081, 9332039515881088707361, 499720579610303128776791, 28785866289100396890228041

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

A007097(n) = Min {k : A109301(k) = n} = the first k whose rote height is n, the level set leader or minimum inverse function corresponding to A109301. - Jon Awbrey, Jun 26 2005
Lubomir Alexandrov informs me that he studied this sequence in his 1965 notebook. - N. J. A. Sloane, May 23 2008
a(n) is the Matula-Goebel number of the rooted path tree on n+1 vertices. The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. - Emeric Deutsch, Feb 18 2012
Conjecture: log(a(1))*log(a(2))*...*log(a(n)) ~ a(n). - Thomas Ordowski, Mar 26 2015

Lubomir Alexandrov, unpublished notes, circa 1960.
L. Longeri, Towards understanding nature and the aesthetics of prime numbers, https://www.longeri.org/prime/nature.html [Broken link, but leave the URL here for historical reasons]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math.NT/9811096, 1998.
Lubomir Alexandrov, "The Eratosthenes Progression p(k+1)=π^{-1}(p(k)), k=0,1,2,..., p(0)=1,4,6,... Determines an Inner Prime Number Distribution Law", Second Int. Conf. "Modern Trends in Computational Physics", Jul 24-29, 2000, Dubna, Russia, Book of Abstracts, p. 19. Available at arXiv:math/0105154 [math.NT], 2001.
Lubomir Alexandrov, Prime Number Sequences And Matrices Generated By Counting Arithmetic Functions, Communications of the Joint Institute of Nuclear Research, E5-2002-55, Dubna, 2002.
J. Awbrey, Riffs and Rotes
R. G. Batchko, A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes, arXiv preprint arXiv:1405.2900 [math.GM], 2014.
Peter R. Cappello, A Note on a Bijection between Natural Numbers and Rooted Trees, 4th SIAM Conference on Discrete Mathematics, June 1988. See section 3 set S codes of paths (codes are per Matula-Goebel).
M. Deléglise, Computation of large values of pi(x)
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992
Index entries for sequences related to Matula-Goebel numbers

Row 1 of array A114537.
Left edge of tree A227413, right edge of A246378.
Cf. A000040, A000720, A049076-A049081, A049084, A057450, A109301, A131842, A006450, A292667.
Cf. A078442, A109082 (left inverses).
Subsequence of A245823.

P:=Filtered([1..60000],IsPrime);;
a:=[1];; for n in [2..10] do a[n]:=P[a[n-1]]; od; a; # Muniru A Asiru, Dec 22 2018

a007097 n = a007097_list !! n
a007097_list = iterate a000040 1  -- Reinhard Zumkeller, Jul 14 2013

seq((ithprime@@n)(1),n=0..10); # Peter Luschny, Oct 16 2012

NestList[Prime@# &, 1, 16] (* Robert G. Wilson v, May 30 2006 *)

print1(p=1);until(,print1(","p=prime(p)))  \\ M. F. Hasler, Oct 09 2011

A049084(a(n+1)) = a(n). - Reinhard Zumkeller, Jul 14 2013
a(n)/a(n-1) ~ log(a(n)) ~ prime(n). - Thomas Ordowski, Mar 26 2015
a(n) = prime^{[n]}(1), with the prime function prime(k) = A000040(k), with a(0) = 1. See the name and the programs. - Wolfdieter Lang, Apr 03 2018
Sum_{n>=1} 1/a(n) = A292667. - Amiram Eldar, Oct 15 2020

a(15) corrected and a(16)-a(17) added by Paul Zimmermann
a(18)-a(19) found by David Baugh using a program by Xavier Gourdon and Andrey V. Kulsha, Oct 25 2007
a(20)-a(21) found by Andrey V. Kulsha using a program by Xavier Gourdon, Oct 02 2011
a(22) from Henri Lifchitz, Oct 14 2014
a(23) from David Baugh using Kim Walisch's primecount, May 16 2016

A006880 Number of primes < 10^n.

Original entry on oeis.org

0, 4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534, 455052511, 4118054813, 37607912018, 346065536839, 3204941750802, 29844570422669, 279238341033925, 2623557157654233, 24739954287740860, 234057667276344607, 2220819602560918840, 21127269486018731928, 201467286689315906290

Offset: 0

N. J. A. Sloane and Simon Plouffe

Number of primes with at most n digits; or pi(10^n).
Partial sums of A006879. - Lekraj Beedassy, Jun 25 2004
Also omega( (10^n)! ), where omega(x): number of distinct prime divisors of x. - Cino Hilliard, Jul 04 2007
This sequence also gives a good approximation for the sum of primes less than 10^(n/2). This is evident from the fact that the number of primes less than 10^2n closely approximates the sum of primes less than 10^n. See link on Sum of Primes for the derivation. - Cino Hilliard, Jun 08 2008
It appears that (10^n)/log((n+3)!) is a lower bound close to a(n), see A025201. - Eric Desbiaux, Jul 20 2010, edited by M. F. Hasler, Dec 03 2018

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 143, 146.
Richard Crandall and Carl B. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; p. 11.
Keith Devlin, Mathematics: The New Golden Age, new and revised edition. New York: Columbia University Press (1993): p. 6, Table 1.
Marcus du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; p. 48.
Calvin T. Long, Elementary Introduction to Number Theory. Prentice-Hall, Englewood Cliffs, NJ, 1987, p. 77.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 179.
H. Riesel, "Prime numbers and computer methods for factorization," Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, page 38.
D. Shanks, Solved and Unsolved Problems in Number Theory. Chelsea, NY, 2nd edition, 1978, p. 15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 455052511 at p. 190.

David Baugh, Table of n, a(n) for n = 0..29 (terms n = 1..27 from Charles R Greathouse IV).
Chris K. Caldwell, How Many Primes Are There?
Chris K. Caldwell, Mark Deleglise's work
Muhammed Hüsrev Cilasun, An Analytical Approach to Exponent-Restricted Multiple Counting Sequences, arXiv preprint arXiv:1412.3265 [math.NT], 2014.
Muhammed Hüsrev Cilasun, Generalized Multiple Counting Jacobsthal Sequences of Fermat Pseudoprimes, Journal of Integer Sequences, Vol. 19, 2016, #16.2.3.
Jens Franke, Thorsten Kleinjung, Jan Büthe, and Alexander Jost, A practical analytic method for calculating pi(x), Math. Comp. 86 (2017), 2889-2909.
Xavier Gourdon, a(22) found by pi(x) project
Xavier Gourdon and Pascal Sebah, The pi(x) project : results and current computations
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Cino Hilliard, Sum of primes [unusable link]
Ronald K. Hoeflin, Titan Test
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979, [see p. 6 of the pdf].
Rishi Kumar, Kepler Sets of Second-Order Linear Recurrence Sequences Over Q_p, arXiv:2406.05890 [math.NT], 2024. See p. 7.
J. C. Lagarias, V. S. Miller, and A. M. Odlyzko, Computing pi(x): The Meissel-Lehmer method, Math. Comp., 44 (1985), pp. 537-560.
J. C. Lagarias and Andrew M. Odlyzko, Computing pi(x): An analytic method, J. Algorithms, 8 (1987), pp. 173-191.
Pieter Moree, Izabela Petrykiewicz, and Alisa Sedunova, A computational history of prime numbers and Riemann zeros, arXiv:1810.05244 [math.NT], 2018. See Table 1 p. 6.
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)
Tomás Oliveira e Silva, Computing pi(x): the combinatorial method, Revista do Detua, Vol. 4, N 6, March 2006.
David J. Platt, Computing pi(x) analytically, arXiv:1203.5712 [math.NT], 2012-2013.
Vladimir Pletser, Conjecture on the value of Pi(10^26), the number of primes less than 10^26, arXiv:1307.4444 [math.NT], 2013.
Vladimir Pletser, Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems, Preprints.org, 2024. See p. 20.
Douglas B. Staple, The combinatorial algorithm for computing pi(x), arXiv:1503.01839 [math.NT], 2015.
M. R. Watkins, The distribution of prime numbers
Eric Weisstein's World of Mathematics, Prime Counting Function
Wikipedia, Prime number theorem
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1989
Index entries for sequences related to numbers of primes in various ranges

Cf. A000720, A006879, A006988, A007053, A011557, A025201, A040014.

a006880 = a000720 . (10 ^)  -- Reinhard Zumkeller, Mar 17 2015

Table[PrimePi[10^n], {n, 0, 14}] (* Jean-François Alcover, Nov 08 2016, corrected Sep 29 2020, a(14) being the maximum computable with certain implementations *)

a(n)=primepi(10^n) \\ Charles R Greathouse IV, Nov 08 2011

a(n) = A000720(10^n). - M. F. Hasler, Dec 03 2018

Limit_{n->oo} a(n)/a(n-1) = 10. - Stefano Spezia, Aug 31 2025

Lehmer gave the incorrect value 455052512 for the 10th term. More terms May 1996. Jud McCranie points out that the 11th term is not 4188054813 but rather 4118054813.
a(22) from Robert G. Wilson v, Sep 04 2001
a(23) (see Gourdon and Sebah) has yet to be verified and the assumed error is +-1. - Robert G. Wilson v, Jul 10 2002 [The actual error was 14037804. - N. J. A. Sloane, Nov 28 2007]
a(23) corrected by N. J. A. Sloane from the web page of Tomás Oliveira e Silva, Nov 28 2007
a(25) from J. Buethe, J. Franke, A. Jost, T. Kleinjung, Jun 01 2013, who said: "We have calculated pi(10^25) = 176846309399143769411680 unconditionally, using an analytic method based on Weil's explicit formula".
a(26) from Douglas B. Staple, Dec 02 2014
a(27) in the b-file from David Baugh and Kim Walisch via Charles R Greathouse IV, Jun 01 2016
a(28) in the b-file from David Baugh and Kim Walisch, Oct 26 2020
a(29) in the b-file from David Baugh and Kim Walisch, Feb 28 2022

A000105 Number of free polyominoes (or square animals) with n cells.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 13079255, 50107909, 192622052, 742624232, 2870671950, 11123060678, 43191857688, 168047007728, 654999700403, 2557227044764, 9999088822075, 39153010938487, 153511100594603

Offset: 0

N. J. A. Sloane

For n>0, a(n) + A030228(n) = A000988(n) because the number of free polyominoes plus the number of polyominoes lacking bilateral symmetry equals the number of one-sided polyominoes. - Graeme McRae, Jan 05 2006
The possible symmetry groups of a (nonempty) polyomino are the 10 subgroups of the dihedral group D_8 of order 8: D_8, 1, Z_2 (five times), Z_4, (Z_2)^2 (twice). - Benoit Jubin, Dec 30 2008
Names for first few polyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, enneomino, decomino, hendecomino, dodecomino, ...
Limit_{n->oo} a(n)^(1/n) = mu with 3.98 < mu < 4.64 (quoted by Castiglione et al., with a reference to Barequet et al., 2006, for the lower bound). The upper bound is due to Klarner and Rivest, 1973. By Madras, 1999, it is now known that this limit, also known as Klarner's constant, is equal to the limit growth rate lim_{n->oo} a(n+1)/a(n).
Polyominoes are worth exploring in the elementary school classroom. Students in grade 2 can reproduce the first 6 terms. Grade 3 students can explore area and perimeter. Grade 4 students can talk about polyomino symmetries.
The pentominoes should be singled out for special attention: 1) they offer a nice, manageable set that a teacher can commercially acquire without too much expense. 2) There are also deeply strategic games and perplexing puzzles that are great for all students. 3) A fraction of students will become engaged because of the beautiful solutions.
Conjecture: Almost all polyominoes are holey. In other words, A000104(n)/a(n) tends to 0 for increasing n. - John Mason, Dec 11 2021 (This is true, a consequence of Madras's 1999 pattern theorem. - Johann Peters, Jan 06 2024)

			a(0) = 1 as there is 1 empty polyomino with #cells = 0. - _Fred Lunnon_, Jun 24 2020

S. W. Golomb, Polyominoes, Appendix D, p. 152; Princeton Univ. Pr. NJ 1994
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 229.
D. A. Klarner, The Mathematical Gardner, p. 252 Wadsworth Int. CA 1981
W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
George E. Martin, Polyominoes - A Guide to Puzzles and Problems in Tiling, The Mathematical Association of America, 1996
Ed Pegg, Jr., Polyform puzzles, in Tribute to a Mathemagician, Peters, 2005, pp. 119-125.
R. C. Read, Some applications of computers in graph theory, in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, pp. 417-444.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Mason, Table of n, a(n) for n = 0..50 (terms 0..45,47 from Toshihiro Shirakawa)
Z. Abel, E. Demaine, M. Demaine, H. Matsui and G. Rote, Common Developments of Several Different Orthogonal Boxes.
G. Barequet, M. Moffie, A. Ribo and G. Rote, Counting polyominoes on twisted cylinders, Integers 6 (2006), A22, 37 pp. (electronic).
K. S. Brown, Polyomino Enumerations
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.
Juris Čerņenoks and Andrejs Cibulis, Tetrads and their Counting, Baltic J. Modern Computing, Vol. 6 (2018), No. 2, 96-106.
A. Clarke, Polyominoes
A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen. 28(1995) 891-904.
I. Jensen, Enumerations of lattice animals and trees, arXiv:cond-mat/0007239 [cond-mat.stat-mech], 2000.
I. Jensen and A. J. Guttmann, Statistics of lattice animals (polyominoes) and polygons, Journal of Physics A: Mathematical and General, vol. 33, pp. L257-L263, 2000.
M. Keller, Counting polyforms.
D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canadian J. of Mathematics, 25 (1973), 585-602.
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Index entries for "core" sequences

Sequences classifying polyominoes by symmetry group: A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.
Cf. A001168 (not reduced by D_8 symmetry), A000104 (no holes), A054359, A054360, A001419, A000988, A030228 (chiral polyominoes).
See A006765 for another version.
Cf. also A000577, A000228, A103465, A210996 (bisection).
Excluding a(0), 8th and 9th row of A366766.

(* In this program by Jaime Rangel-Mondragón, polyominoes are represented as a list of Gaussian integers. *)
polyominoQ[p_List] := And @@ ((IntegerQ[Re[#]] && IntegerQ[Im[#]])& /@ p);
rot[p_?polyominoQ] := I*p;
ref[p_?polyominoQ] := (# - 2 Re[#])& /@ p;
cyclic[p_] := Module[{i = p, ans = {p}}, While[(i = rot[i]) != p, AppendTo[ans, i]]; ans];
dihedral[p_?polyominoQ] := Flatten[{#, ref[#]}& /@ cyclic[p], 1];
canonical[p_?polyominoQ] := Union[(# - (Min[Re[p]] + Min[Im[p]]*I))& /@ p];
allPieces[p_] := Union[canonical /@ dihedral[p]];
polyominoes[1] = {{0}};
polyominoes[n_] := polyominoes[n] = Module[{f, fig, ans = {}}, fig = ((f = #1; ({f, #1 + 1, f, #1 + I, f, #1 - 1, f, #1 - I}&) /@ f)&) /@ polyominoes[n - 1]; fig = Partition[Flatten[fig], n]; f = Select[Union[canonical /@ fig], Length[#1] == n &]; While[f != {}, ans = {ans, First[f]}; f = Complement[f, allPieces[First[f]]]]; Partition[Flatten[ans], n]];
a[n_] := a[n] = Length[ polyominoes[n]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 12}] (* Jean-François Alcover, Mar 24 2015, after Jaime Rangel-Mondragón *)

a(n) = A000104(n) + A001419(n). - R. J. Mathar, Jun 15 2014
a(n) = A006749(n) + A006746(n) + A006748(n) + A006747(n) + A056877(n) + A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018
a(n) = A259087(n) + A259088(n). - R. J. Mathar, May 22 2019
a(n) = (4*A006746(n) + 4*A006748(n) + 4*A006747(n) + 6*A056877(n) + 6*A056878(n) + 6*A144553(n) + 7*A142886(n) + A001168(n))/8. - John Mason, Nov 14 2021

Extended to n=28 by Tomás Oliveira e Silva
Link updated by William Rex Marshall, Dec 16 2009
Edited by Gill Barequet, May 24 2011
Misspelling "polyominos" corrected by Don Knuth, May 03 2016
a(29)-a(45), a(47) from Toshihiro Shirakawa
a(46) calculated using values from A001168 (I. Jensen), A006748/A056877/A056878/A144553/A142886 (Robert A. Russell) and A006746/A006747 (John Mason), Nov 14 2021

A001220 Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.

Original entry on oeis.org

1093, 3511

Offset: 1

N. J. A. Sloane

Sequence is believed to be infinite.
Joseph Silverman showed that the abc-conjecture implies that there are infinitely many primes which are not in the sequence. - Benoit Cloitre, Jan 09 2003
Graves and Murty (2013) improved Silverman's result by showing that for any fixed k > 1, the abc-conjecture implies that there are infinitely many primes == 1 (mod k) which are not in the sequence. - Jonathan Sondow, Jan 21 2013
The squares of these numbers are Fermat pseudoprimes to base 2 (A001567) and Catalan pseudoprimes (A163209). - T. D. Noe, May 22 2003
Primes p that divide the numerator of the harmonic number H((p-1)/2); that is, p divides A001008((p-1)/2). - T. D. Noe, Mar 31 2004
In a 1977 paper, Wells Johnson, citing a suggestion from Lawrence Washington, pointed out the repetitions in the binary representations of the numbers which are one less than the two known Wieferich primes; i.e., 1092 = 10001000100 (base 2); 3510 = 110110110110 (base 2). It is perhaps worth remarking that 1092 = 444 (base 16) and 3510 = 6666 (base 8), so that these numbers are small multiples of repunits in the respective bases. Whether this is mathematically significant does not appear to be known. - John Blythe Dobson, Sep 29 2007
A002326((a(n)^2 - 1)/2) = A002326((a(n)-1)/2). - Vladimir Shevelev, Jul 09 2008, Aug 24 2008
It is believed that p^2 does not divide 3^(p-1) - 1 if p = a(n). This is true for n = 1 and 2. See A178815, A178844, A178900, and Ostafe-Shparlinski (2010) Section 1.1. - Jonathan Sondow, Jun 29 2010
These primes also divide the numerator of the harmonic number H(floor((p-1)/4)). - H. Eskandari (hamid.r.eskandari(AT)gmail.com), Sep 28 2010
1093 and 3511 are prime numbers p satisfying congruence 429327^(p-1) == 1 (mod p^2). Why? - Arkadiusz Wesolowski, Apr 07 2011. Such bases are listed in A247208. - Max Alekseyev, Nov 25 2014. See A269798 for all such bases, prime and composite, that are not powers of 2. - Felix Fröhlich, Apr 07 2018
A196202(A049084(a(1))) = A196202(A049084(a(2))) = 1. - Reinhard Zumkeller, Sep 29 2011
If q is prime and q^2 divides a prime-exponent Mersenne number, then q must be a Wieferich prime. Neither of the two known Wieferich primes divide Mersenne numbers. See Will Edgington's Mersenne page in the links below. - Daran Gill, Apr 04 2013
There are no other terms below 4.97*10^17 as established by PrimeGrid (see link below). - Max Alekseyev, Nov 20 2015. The search was done via PrimeGrid's PRPNet and the results were not double-checked. Because of the unreliability of the testing, the search was suspended in May 2017 (cf. Goetz, 2017). - Felix Fröhlich, Apr 01 2018. On Nov 28 2020, PrimeGrid has resumed the search (cf. Reggie, 2020). - Felix Fröhlich, Nov 29 2020. As of Dec 29 2022, PrimeGrid has completed the search to 2^64 (about 1.8 * 10^19) and has no plans to continue further. - Charles R Greathouse IV, Sep 24 2024
Are there other primes q >= p such that q^2 divides 2^(p-1)-1, where p is a prime? - Thomas Ordowski, Nov 22 2014. Any such q must be a Wieferich prime. - Max Alekseyev, Nov 25 2014
Primes p such that p^2 divides 2^r - 1 for some r, 0 < r < p. - Thomas Ordowski, Nov 28 2014, corrected by Max Alekseyev, Nov 28 2014
For some reason, both p=a(1) and p=a(2) also have more bases b with 1 < b < p that make b^(p-1) == 1 (mod p^2) than any smaller prime p; in other words, a(1) and a(2) belong to A248865. - Jeppe Stig Nielsen, Jul 28 2015
Let r_1, r_2, r_3, ..., r_i be the set of roots of the polynomial X^((p-1)/2) - (p-3)! * X^((p-3)/2) - (p-5)! * X^((p-5)/2) - ... - 1. Then p is a Wieferich prime iff p divides sum{k=1, p}(r_k^((p-1)/2)) (see Example 2 in Jakubec, 1994). - Felix Fröhlich, May 27 2016
Arthur Wieferich showed that if p is not a term of this sequence, then the First Case of Fermat's Last Theorem has no solution in x, y and z for prime exponent p (cf. Wieferich, 1909). - Felix Fröhlich, May 27 2016
Let U_n(P, Q) be a Lucas sequence of the first kind, let e be the Legendre symbol (D/p) and let p be a prime not dividing 2QD, where D = P^2 - 4*Q. Then a prime p such that U_(p-e) == 0 (mod p^2) is called a "Lucas-Wieferich prime associated to the pair (P, Q)". Wieferich primes are those Lucas-Wieferich primes that are associated to the pair (3, 2) (cf. McIntosh, Roettger, 2007, p. 2088). - Felix Fröhlich, May 27 2016
Any repeated prime factor of a term of A000215 is a term of this sequence. Thus, if there exist infinitely many Fermat numbers that are not squarefree, then this sequence is infinite, since no two Fermat numbers share a common factor. - Felix Fröhlich, May 27 2016
If the Diophantine equation p^x - 2^y = d has more than one solution in positive integers (x, y), with (p, d) not being one of the pairs (3, 1), (3, -5), (3, -13) or (5, -3), then p is a term of this sequence (cf. Scott, Styer, 2004, Corollary to Theorem 2). - Felix Fröhlich, Jun 18 2016
Odd primes p such that Chi_(D_0)(p) != 1 and Lambda_p(Q(sqrt(D_0))) != 1, where D_0 < 0 is the fundamental discriminant of the imaginary quadratic field Q(sqrt(1-p^2)) and Chi and Lambda are Iwasawa invariants (cf. Byeon, 2006, Proposition 1 (i)). - Felix Fröhlich, Jun 25 2016
If q is an odd prime, k, p are primes with p = 2*k+1, k == 3 (mod 4), p == -1 (mod q) and p =/= -1 (mod q^3) (Jakubec, 1998, Corollary 2 gives p == -5 (mod q) and p =/= -5 (mod q^3)) with the multiplicative order of q modulo k = (k-1)/2 and q dividing the class number of the real cyclotomic field Q(Zeta_p + (Zeta_p)^(-1)), then q is a term of this sequence (cf. Jakubec, 1995, Theorem 1). - Felix Fröhlich, Jun 25 2016
From Felix Fröhlich, Aug 06 2016: (Start)
Primes p such that p-1 is in A240719.
Prime terms of A077816 (cf. Agoh, Dilcher, Skula, 1997, Corollary 5.9).
p = prime(n) is in the sequence iff T(2, n) > 1, where T = A258045.
p = prime(n) is in the sequence iff an integer k exists such that T(n, k) = 2, where T = A258787. (End)
Conjecture: an integer n > 1 such that n^2 divides 2^(n-1)-1 must be a Wieferich prime. - Thomas Ordowski, Dec 21 2016
The above conjecture is equivalent to the statement that no "Wieferich pseudoprimes" (WPSPs) exist. While base-b WPSPs are known to exist for several bases b > 1 other than 2 (see for example A244752), no base-2 WPSPs are known. Since two necessary conditions for a composite to be a base-2 WPSP are that, both, it is a base-2 Fermat pseudoprime (A001567) and all its prime factors are Wieferich primes (cf. A270833), as shown in the comments in A240719, it seems that the first base-2 WPSP, if it exists, is probably very large. This appears to be supported by the guess that the properties of a composite to be a term of A001567 and of A270833 are "independent" of each other and by the observation that the scatterplot of A256517 seems to become "less dense" at the x-axis parallel line y = 2 for increasing n. It has been suggested in the literature that there could be asymptotically about log(log(x)) Wieferich primes below some number x, which is a function that grows to infinity, but does so very slowly. Considering the above constraints, the number of WPSPs may grow even more slowly, suggesting any such number, should it exist, probably lies far beyond any bound a brute-force search could reach in the forseeable future. Therefore I guess that the conjecture may be false, but a disproof or the discovery of a counterexample are probably extraordinarily difficult problems. - Felix Fröhlich, Jan 18 2019
Named after the German mathematician Arthur Josef Alwin Wieferich (1884-1954). a(1) = 1093 was found by Waldemar Meissner in 1913. a(2) = 3511 was found by N. G. W. H. Beeger in 1922. - Amiram Eldar, Jun 05 2021
From Jianing Song, Jun 21 2025: (Start)
The ring of integers of Q(2^(1/k)) is Z[2^(1/k)] if and only if k does not have a prime factor in this sequence (k is in A342390). See Theorem 5.3 of the paper of Keith Conrad. For example, we have:
  (1 + 2^(364/1093) + 2^(2*364/1093) + ... + 2^(1092*364/1093))/1093 is an algebraic integer, but it is not in Z[2^(1/1093)];
  (1 + 2^(1755/3511) + 2^(2*1755/3511) + ... + 2^(3510*1755/3511))/3511 is an algebraic integer, but it is not in Z[2^(1/3511)]. (End)

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Cf. similar primes related to the first case of Fermat's last theorem: A007540, A088164.
Sequences "primes p such that p^2 divides X^(p-1)-1": A014127 (X=3), A123692 (X=5), A212583 (X=6), A123693 (X=7), A045616 (X=10), A111027 (X=12), A128667 (X=13), A234810 (X=14), A242741 (X=15), A128668 (X=17), A244260 (X=18), A090968 (X=19), A242982 (X=20), A298951 (X=22), A128669 (X=23), A306255 (X=26), A306256 (X=30).
Cf. A001567, A002323, A077816, A001008, A039951, A049094, A126196, A126197, A178815, A178844, A178871, A178900, A246503, A247208, A269798.

Filtered([1..50000],p->IsPrime(p) and (2^(p-1)-1) mod p^2 =0); # Muniru A Asiru, Apr 03 2018

import Data.List (elemIndices)
a001220 n = a001220_list !! (n-1)
a001220_list = map (a000040 . (+ 1)) $ elemIndices 1 a196202_list
-- Reinhard Zumkeller, Sep 29 2011

[p : p in PrimesUpTo(310000) | IsZero((2^(p-1) - 1) mod (p^2))]; // Vincenzo Librandi, Jan 19 2019

wieferich := proc (n) local nsq, remain, bin, char: if (not isprime(n)) then RETURN("not prime") fi: nsq := n^2: remain := 2: bin := convert(convert(n-1, binary),string): remain := (remain * 2) mod nsq: bin := substring(bin,2..length(bin)): while (length(bin) > 1) do: char := substring(bin,1..1): if char = "1" then remain := (remain * 2) mod nsq fi: remain := (remain^2) mod nsq: bin := substring(bin,2..length(bin)): od: if (bin = "1") then remain := (remain * 2) mod nsq fi: if remain = 1 then RETURN ("Wieferich prime") fi: RETURN ("non-Wieferich prime"): end: # Ulrich Schimke (ulrschimke(AT)aol.com), Nov 01 2001

Select[Prime[Range[50000]],Divisible[2^(#-1)-1,#^2]&]  (* Harvey P. Dale, Apr 23 2011 *)
Select[Prime[Range[50000]],PowerMod[2,#-1,#^2]==1&] (* Harvey P. Dale, May 25 2016 *)

N=10^4; default(primelimit,N);
forprime(n=2,N,if(Mod(2,n^2)^(n-1)==1,print1(n,", ")));
\\ Joerg Arndt, May 01 2013

from sympy import prime
from gmpy2 import powmod
A001220_list = [p for p in (prime(n) for n in range(1,10**7)) if powmod(2,p-1,p*p) == 1]
# Chai Wah Wu, Dec 03 2014

(A178815(A000720(p))^(p-1) - 1) mod p^2 = A178900(n), where p = a(n). - Jonathan Sondow, Jun 29 2010

Odd primes p such that A002326((p^2-1)/2) = A002326((p-1)/2). See A182297. - Thomas Ordowski, Feb 04 2014

A000001 Number of groups of order n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, 1, 10, 1, 4, 2

Offset: 0

N. J. A. Sloane

Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - Lekraj Beedassy, Dec 16 2004
Also, number of nonisomorphic primitives (antiderivatives) of the combinatorial species Lin[n-1], or X^{n-1}; see Rajan, Summary item (i). - Nicolae Boicu, Apr 29 2011
In (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), a(n) is called the "group number of n", denoted by gnu(n), and the first occurrence of k is called the "minimal order attaining k", denoted by moa(k) (see A046057). - Daniel Forgues, Feb 15 2017
It is conjectured in (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008) that the sequence n -> a(n) -> a(a(n)) = a^2(n) -> a(a(a(n))) = a^3(n) -> ... -> consists ultimately of 1s, where a(n), denoted by gnu(n), is called the "group number of n". - Muniru A Asiru, Nov 19 2017
MacHale (2020) shows that there are infinitely many values of n for which there are more groups than rings of that order (cf. A027623). He gives n = 36355 as an example. It would be nice to have enough values of n to create an OEIS entry for them. - N. J. A. Sloane, Jan 02 2021
I conjecture that a(i) * a(j) <= a(i*j) for all nonnegative integers i and j. - Jorge R. F. F. Lopes, Apr 21 2024

			Groups of orders 1 through 10 (C_n = cyclic, D_n = dihedral of order n, Q_8 = quaternion, S_n = symmetric):
1: C_1
2: C_2
3: C_3
4: C_4, C_2 X C_2
5: C_5
6: C_6, S_3=D_6
7: C_7
8: C_8, C_4 X C_2, C_2 X C_2 X C_2, D_8, Q_8
9: C_9, C_3 X C_3
10: C_10, D_10

S. R. Blackburn, P. M. Neumann, and G. Venkataraman, Enumeration of Finite Groups, Cambridge, 2007.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 302, #35.
J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 150.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley Publ. Co., Reading, MA, 1989, Section 6.6 'Fibonacci Numbers' pp. 281-283.
M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
D. Joyner, 'Adventures in Group Theory', Johns Hopkins Press. Pp. 169-172 has table of groups of orders < 26.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.
M. F. Newman and E. A. O'Brien, A CAYLEY library for the groups of order dividing 128. Group theory (Singapore, 1987), 437-442, de Gruyter, Berlin-New York, 1989.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. U. Besche and Ivan Panchenko, Table of n, a(n) for n = 0..2047 [Terms 1 through 2015 copied from Small Groups Library mentioned below. Terms 2016 - 2047 added by _Ivan Panchenko_, Aug 29 2009. 0 prepended by _Ray Chandler_, Sep 16 2015. a(1024) corrected by _Benjamin Przybocki_, Jan 06 2022]
H. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927).
Hans Ulrich Besche and Bettina Eick, Construction of finite groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
Hans Ulrich Besche and Bettina Eick, The groups of order at most 1000 except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
H. U. Besche, B. Eick and E. A. O'Brien, The groups of order at most 2000, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.
H. U. Besche, B. Eick, E. A. O'Brien and Max Horn, The Small Groups Library
H. U. Besche, B. Eick and E. A. O'Brien, Number of isomorphism types of finite groups of given order [gives incorrect a(1024)]
H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
Henry Bottomley, Illustration of initial terms
David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3.
J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.
Marek Dančo, Mikoláš Janota, Michael Codish, and João Jorge Araújo, Complete Symmetry Breaking for Finite Models, arXiv:2502.10155 [cs.LO], 2025. See p. 7. Mentions this sequence.
Yang-Hui He and Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019.
Otto Hölder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 43 pp. 301-412 (1893).
Max Horn, Numbers of isomorphism types of finite groups of given order
Rodney James, The groups of order p^6 (p an odd prime), Math. Comp. 34 (1980), 613-637.
Rodney James and John Cannon, Computation of isomorphism classes of p-groups, Mathematics of Computation 23.105 (1969): 135-140.
Olexandr Konovalov, Crowdsourcing project for the database of numbers of isomorphism types of finite groups, Github (a list of gnu(n) for many n < 50000).
Desmond MacHale, Are There More Finite Rings than Finite Groups?, Amer. Math. Monthly (2020) Vol. 127, Issue 10, 936-938.
Mehdi Makhul, Josef Schicho, and Audie Warren, On Galois groups of type-1 minimally rigid graphs, arXiv:2306.04392 [math.CO], 2023.
G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Dayanand S. Rajan, The equations D^kY=X^n in combinatorial species, Discrete Mathematics 118 (1993) 197-206 North-Holland.
E. Rodemich, The groups of order 128, J. Algebra 67 (1980), no. 1, 129-142.
Gordon Royle, Combinatorial Catalogues. See subpage "Generators of small groups" for explicit generators for most groups of even order < 1000. [broken link]
D. Rusin, Asymptotics [Cached copy of lost web page]
Eric Weisstein's World of Mathematics, Finite Group
Wikipedia, Finite group
M. Wild, The groups of order sixteen made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
Gang Xiao, SmallGroup
Index entries for sequences related to groups
Index entries for "core" sequences

The main sequences concerned with group theory are A000001 (this one), A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A000688, A060689, A051532.
Cf. A046058, A046059, A023675, A023676.
A003277 gives n for which A000001(n) = 1, A063756 (partial sums).
A046057 gives first occurrence of each k.
A027623 gives the number of rings of order n.

A000001 := Concatenation([0], List([1..500], n -> NumberSmallGroups(n))); # Muniru A Asiru, Oct 15 2017

D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] ]; // John Cannon, Dec 23 2006

GroupTheory:-NumGroups(n); # with(GroupTheory); loads this command - N. J. A. Sloane, Dec 28 2017

FiniteGroupCount[Range[100]] (* Harvey P. Dale, Jan 29 2013 *)
a[ n_] := If[ n < 1, 0, FiniteGroupCount @ n]; (* Michael Somos, May 28 2014 *)

From Mitch Harris, Oct 25 2006: (Start)
For p, q, r primes:
a(p) = 1, a(p^2) = 2, a(p^3) = 5, a(p^4) = 14, if p = 2, otherwise 15.
a(p^5) = 61 + 2*p + 2*gcd(p-1,3) + gcd(p-1,4), p >= 5, a(2^5)=51, a(3^5)=67.
a(p^e) ~ p^((2/27)e^3 + O(e^(8/3))).
a(p*q) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p. (p < q)
a(p*q^2) is one of the following:
   ---------------------------------------------------------------------------
  | a(p*q^2) |            p*q^2 of the form             |  Sequences (p*q^2)  |
   ---------- ------------------------------------------ ---------------------
  | (p+9)/2  |          q == 1 (mod p), p odd           |      A350638        |
  |    5     |         p=3, q=2 =>  p*q^2 = 12          |Special case with A_4|
  |    5     |               p=2, q odd                 |      A143928        |
  |    5     |             p == 1 (mod q^2)             |      A350115        |
  |    4     | p == 1 (mod q), p > 3, p !== 1 (mod q^2) |      A349495        |
  |    3     |       q == -1 (mod p), p and q odd       |      A350245        |
  |    2     |  q !== +-1 (mod p) and p !== 1 (mod q)   |      A350422        |
   ---------------------------------------------------------------------------
[Table from Bernard Schott, Jan 18 2022]
a(p*q*r) (p < q < r) is one of the following:
  q == 1 (mod p)  r == 1 (mod p)  r == 1 (mod q)  a(p*q*r)
  --------------  --------------  --------------  --------
        No              No              No            1
        No              No              Yes           2
        No              Yes             No            2
        No              Yes             Yes           4
        Yes             No              No            2
        Yes             No              Yes           3
        Yes             Yes             No           p+2
        Yes             Yes             Yes          p+4
[table from Derek Holt].
(End)
a(n) = A000688(n) + A060689(n). - R. J. Mathar, Mar 14 2015

More terms from Michael Somos

Typo in b-file description fixed by David Applegate, Sep 05 2009

A006126 Number of hierarchical models on n labeled factors or variables with linear terms forced. Also number of antichain covers of a labeled n-set.

Original entry on oeis.org

2, 1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966, 286386577668298410623295216696338374471993

Offset: 0

N. J. A. Sloane

An antichain cover is a cover such that no element of the cover is a subset of another element of the cover.
Also, the number of nondegenerate monotone Boolean functions of n variables in an n-variable Boolean algebra. - Rodrigo A. Obando (R.Obando(AT)computer.org), Jul 26 2004
Also, number of simplicial complexes on an n-element vertex set. - Richard Stanley, Feb 10 2019
There are two antichains of size zero, namely {} and {{}}, while there is only one simplicial complex, namely {}. The unlabeled case is A006602. The non-covering case is A000372, which is A014466 plus 1. - Gus Wiseman, Mar 31 2019
From Petros Hadjicostas, Apr 10 2020: (Start)
Hierarchical models are always nonempty because they always include an intercept (or overall effect).
The total number of log-linear hierarchical models on n labeled factors (categorical variables) with no forcing of terms is given by A000372(n) - 1 (Dedekind numbers minus 1).
Hierarchical log-linear models for analyzing contingency tables are defined in the classic book by Bishop, Fienberg, and Holland (1975). (End)

			a(5) = 1 + 90 + 790 + 1895 + 2116 + 1375 + 490 + 115 + 20 + 2 = 6894.
There are 9 antichain covers of a labeled 3-set: {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}, {{1},{2},{3}}, {{1,2},{1,3},{2,3}}.
From _Gus Wiseman_, Feb 23 2019: (Start)
The a(0) = 2 through a(3) = 9 antichains:
  {}    {{1}}  {{12}}    {{123}}
  {{}}         {{1}{2}}  {{1}{23}}
                         {{2}{13}}
                         {{3}{12}}
                         {{12}{13}}
                         {{12}{23}}
                         {{13}{23}}
                         {{1}{2}{3}}
                         {{12}{13}{23}}
(End)

Y. M. M. Bishop, S. E. Fienberg and P. W. Holland, Discrete Multivariate Analysis. MIT Press, 1975, p. 34. [In part (e), the Hierarchy Principle for log-linear models is defined. It essentially says that if a higher-order parameter term is included in the log-linear model, then all the lower-order parameter terms should also be included. - Petros Hadjicostas, Apr 08 2020]
V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
C. L. Mallows, personal communication.
A. A. Mcintosh, personal communication.
R. A. Obando, On the number of nondegenerate monotone boolean functions of n variables, In Preparation.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. Baumann and H. Strass, On the number of bipolar Boolean functions, 2014, preprint.
R. Baumann and H. Strass, On the number of bipolar Boolean functions, Journal of Logic and Computation, 27(8) (2017), 2431-2449.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See p. 2.
Florian Bridoux, Amélia Durbec, Kévin Perrot, and Adrien Richard, Complexity of fixed point counting problems in Boolean Networks, arXiv:2012.02513 [math.CO], 2020.
Florian Bridoux, Nicolas Durbec, Kevin Perrot, and Adrien Richard, Complexity of Maximum Fixed Point Problem in Boolean Networks, Conference on Computability in Europe (CiE 2019) Computing with Foresight and Industry (Lecture Notes in Computer Science book series, Vol. 11558), Springer, Cham, 132-143.
K. S. Brown, Dedekind's problem
Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991
C. L. Mallows & N. J. A. Sloane, Emails, May 1991
C. L. Mallows & N. J. A. Sloane, Emails, Jun. 1991
Eric Weisstein's World of Mathematics, Antichain.
Eric Weisstein's World of Mathematics, Cover.
R. I. P. Wickramasinghe, Topics in log-linear models, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008. [From the A000372(2) - 1 = 4 hierarchical log-linear models on two factors X and Y, on p. 18 of his thesis, only Models 11 and 15 force all the linear terms (i.e., a(2) = 2). From the A000372(3) - 1 = 19 hierarchical log-linear models on three factors X, Y, and Z, on p. 36 of his thesis, only Models 11-19 force all the linear terms (i.e., a(3) = 9). - _Petros Hadjicostas_, Apr 08 2020]
D. H. Wiedemann, Letter to N. J. A. Sloane, Nov 03, 1990
D. H. Wiedermann, Email to N. J. A. Sloane, May 28 1991
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A000372, A056046-A056049, A056052, A056101, A056104, A051112-A051118.

Cf. A006602, A014466, A261005, A293606, A293993, A305000, A305844, A306550, A307249, A317674, A319721, A320449.

nn=4;
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n]],SubsetQ],Union@@#==Range[n]&]],{n,0,nn}] (* Gus Wiseman, Feb 23 2019 *)
A000372 = Cases[Import["https://oeis.org/A000372/b000372.txt", "Table"], {, }][[All, 2]];
lg = Length[A000372];
a372[n_] := If[0 <= n <= lg-1, A000372[[n+1]], 0];
a[n_] := Sum[(-1)^(n-k+1) Binomial[n, k-1] a372[k-1], {k, 0, lg}];
a /@ Range[0, lg-1] (* Jean-François Alcover, Jan 07 2020 *)

a(n) = Sum_{k = 1..C(n, floor(n/2))} b(k, n), where b(k, n) is the number of k-antichain covers of a labeled n-set.

Inverse binomial transform of A000372. - Gus Wiseman, Feb 24 2019

Last 3 terms from Michael Bulmer (mrb(AT)maths.uq.edu.au)
Antichain interpretation from Vladeta Jovovic and Goran Kilibarda, Jul 31 2000
a(0) = 2 added by Gus Wiseman, Feb 23 2019
Name edited by Petros Hadjicostas, Apr 08 2020
a(9) using A000372 added by Bruno L. O. Andreotti, May 14 2023

cp's OEIS Frontend

A000043 Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.

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A000668 Mersenne primes (primes of the form 2^n - 1).

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A056654 Numbers k such that 10*R_k + 3 is prime, where R_k is the repunit (A002275) of length k.

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A019434 Fermat primes: primes of the form 2^(2^k) + 1, for some k >= 0.

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A007097 Primeth recurrence: a(n+1) = a(n)-th prime.

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A006880 Number of primes < 10^n.

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