A135650 -id:A135650 - 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

A090748 Numbers k such that 2^(k+1) - 1 is prime.

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 18, 30, 60, 88, 106, 126, 520, 606, 1278, 2202, 2280, 3216, 4252, 4422, 9688, 9940, 11212, 19936, 21700, 23208, 44496, 86242, 110502, 132048, 216090, 756838, 859432, 1257786, 1398268, 2976220, 3021376, 6972592, 13466916, 20996010, 24036582, 25964950, 30402456, 32582656

Offset: 1

Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 03 2004

nonn

Perfect numbers A000396(n) = 2^A133033(n) - 2^a(n), assuming there are no odd perfect numbers. - Omar E. Pol, Feb 24 2008
Number of proper divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n). - Omar E. Pol, Feb 28 2008
Base 2 logarithm of n-th even superperfect number A061652(n). Also base 2 logarithm of n-th superperfect number A019279(n), assuming there are no odd superperfect numbers. - Omar E. Pol, Apr 11 2008
Number of 0's in binary expansion of n-th even perfect number (see A135650). - Omar E. Pol, May 04 2008

			1 is in the sequence because 2^2 - 1 = 3 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Ivan Panchenko)
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

a(n) = A000043(n) - 1. A000043 is the main entry for this sequence.

Cf. A000396, A133033, A000668, A019279, A061652, A135650, A051027.

[n: n in [1..5*10^3] |IsPrime(2^(n+1)-1)]; // Vincenzo Librandi, Jul 28 2016

Select[Range[0, 10^4], PrimeQ[2^(# + 1) - 1] &] (* Vincenzo Librandi, Jul 28 2016 *)

is(n)=ispseudoprime(2^(n+1)-1) \\ Charles R Greathouse IV, Aug 21 2016

a(n) = A000043(n) - 1.

2^(a(n) + 1) = A051027(2^a(n)). - Juri-Stepan Gerasimov, Aug 21 2016 [corrected by Jerzy R Borysowicz, Feb 26 2025]

Edited, corrected and extended by Robert G. Wilson v, Feb 09 2004
a(39) from Omar E. Pol, Jan 23 2009
a(40)-a(44) from Ivan Panchenko, Apr 11 2018

A133028 Even perfect numbers divided by 2.

Original entry on oeis.org

3, 14, 248, 4064, 16775168, 4294934528, 68719345664, 1152921504069976064, 1329227995784915872327346307976921088, 95780971304118053647396689042151819065498660774084608, 6582018229284824168619876730229361455111736159193471558891864064, 7237005577332262213973186563042994240786838745737417944533177174565599576064

Offset: 1

Omar E. Pol, Oct 20 2007, Apr 23 2008, Apr 28 2009

nonn

a(13) has 314 digits and is too large to include. - R. J. Mathar, Oct 23 2007
Largest proper divisor of n-th even perfect number.
Also numbers k such that A000203(k) is divisible 24. - Ctibor O. Zizka, Jun 29 2009

Amiram Eldar, Table of n, a(n) for n = 1..15
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A028334, A000396 (perfect numbers).
Cf. A000043, A000668, A000203.
Cf. A018254, A018487, A032742, A133024, A133025, A135652, A135653, A135654, A135655.
Cf. A134708, A135650. - Omar E. Pol, Jul 07 2009

a:=proc(n) if isprime(2^n-1)=true then 2^(n-2)*(2^n-1) else end if end proc: seq(a(n),n=1..120); # Emeric Deutsch, Oct 24 2007

p = Select[2^Range[400] - 1, PrimeQ]; p*(p+1)/4 (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
Map[2^(#-2) * (2^# - 1) &, MersennePrimeExponent[Range[12]]] (* Amiram Eldar, Oct 21 2024 *)

a(n) = A000396(n)/2. - R. J. Mathar, Oct 23 2007 [Assuming there are no odd perfect numbers. - Jianing Song, Sep 17 2022]
a(n) = 2^(A000043(n) - 2) * A000668(n). - Omar E. Pol, Mar 01 2008
a(n) = A032742(A000396(n)), assuming there are no odd perfect numbers.

More terms from R. J. Mathar and Emeric Deutsch, Oct 23 2007

A133033 Number of proper divisors of n-th even perfect number.

Original entry on oeis.org

3, 5, 9, 13, 25, 33, 37, 61, 121, 177, 213, 253, 1041, 1213, 2557, 4405, 4561, 6433, 8505, 8845, 19377, 19881, 22425, 39873, 43401, 46417, 88993, 172485, 221005, 264097, 432181, 1513677, 1718865, 2515573, 2796537, 5952441, 6042753, 13945185, 26933833, 41992021, 48073165, 51929901, 60804913

Offset: 1

Omar E. Pol, Oct 27 2007, Feb 23 2008, Apr 28 2009

nonn

Perfect numbers: A000396(n) = 2^a(n) - 2^A090748(n), assuming there are no odd perfect numbers.
Also, a(n) is equal to the number of bits in A135650(n), the n-th even perfect number written in base 2.
These are odd numbers k such that 2^((k+1)/2) - 1 is (a Mersenne) prime. - Thomas Ordowski, Apr 20 2019

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Ivan Panchenko)
Chris K. Caldwell, "Top Twenty" page, Mersenne Primes.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A061645, A090748, A135650.

2 * MersennePrimeExponent[Range[48]] - 1 (* Amiram Eldar, Oct 18 2024 *)

a(n) = A061645(n) - 1.

a(n) = A000043(n) + A090748(n) = 2*A000043(n) - 1 = 2*A090748(n) + 1.

a(39)-a(43) from Ivan Panchenko, Apr 12 2018

A160962 Perfect numbers written in base 4.

Original entry on oeis.org

12, 130, 13300, 1333000, 1333333000000, 13333333300000000, 1333333333000000000, 1333333333333333000000000000000, 1333333333333333333333333333333000000000000000000000000000000

Offset: 1

Vladislav-Stepan Malakhovsky and Juri-Stepan Gerasimov, May 31 2009

Amiram Eldar, Table of n, a(n) for n = 1..14

Cf. A000043, A000396, A000668, A135650, A161322.

Map[FromDigits[IntegerDigits[#, 4]] &, PerfectNumber[Range[10]]] (* Amiram Eldar, Oct 16 2024 *)

a(n) = A007090(A000396(n)). - Amiram Eldar, Oct 16 2024

Edited by N. J. A. Sloane, Jun 07 2009

A161323 Perfect numbers written in base 8.

Original entry on oeis.org

6, 34, 760, 17700, 177770000, 77777600000, 1777777000000, 177777777770000000000, 17777777777777777777700000000000000000000, 77777777777777777777777777777600000000000000000000000000000

Offset: 1

Juri-Stepan Gerasimov, Jun 07 2009

All members end in 0, 1, 4, 5, or 6. Conjecturally, apart from the first two terms, all terms end in 0.

Harvey P. Dale, Table of n, a(n) for n = 1..15

Cf. A000396, A135650, A160962.

FromDigits[IntegerDigits[#,8]]&/@PerfectNumber[Range[10]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 08 2019 *)

Edited by Charles R Greathouse IV, Mar 19 2010

A135651 Even superperfect numbers written in base 2.

Original entry on oeis.org

10, 100, 10000, 1000000, 1000000000000, 10000000000000000, 1000000000000000000, 1000000000000000000000000000000

Offset: 1

Omar E. Pol, Feb 23 2008

Also, superperfect numbers (A019279) written in base 2 (If there are no odd perfect numbers).
Also, concatenation of "1" and A090748(n) digits "0".
The number of digits of a(n) is equal to A000043(n) and also equal to the number of digits of n-th Mersenne prime written in base 2 (see A117293, A135650).

			a(3)=10000 because the 3rd even superperfect number A061652(3)=16 and 16 written in base 2 is equal to 10000.

Cf. A000043, A000668, A019279, A061652, A090748, A117293, A135650.

A135656 Perfect numbers divided by 2, written in base 2.

Original entry on oeis.org

11, 1110, 11111000, 111111100000, 111111111111100000000000, 11111111111111111000000000000000, 111111111111111111100000000000000000, 111111111111111111111111111111100000000000000000000000000000

Offset: 1

Omar E. Pol, Feb 28 2008

The number of divisors of a(n) is equal to the number of its digits. This number is equal to 2*A000043(n)-2. The number of divisors of a(n) that are powers of 2 is equal to the number of divisors that are multiples of n-th Mersenne prime A000668(n) and this number of divisors is equal to A090748(n). The first digits of a(n) are "1". For n>1 the last digits are "0". The number of digits "1" is equal to A000043(n). The number of digits "0" is equal to A000043(n)-2. The concatenation of digits "1" gives the n-th Mersenne prime written in binary (see A117293(n)). The structure of divisors of a(n) represent a triangle (see example).

			a(4)=111111100000 because the 4th. perfect number is 8128 and 8128/2=4064 and 4064 written in base 2 is 111111100000. Note that 1111111 is the 4th. Mersenne prime A000668(4)=127, written in base 2.
The structure of divisors of a(4)=111111100000

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Perfect numbers divided by 2: A133028. Cf. A000396, A000668, A019279, A090748, A117293, A135650.

a(n)=A133028(n) written in base 2.

A161496 Number of zeros in n-th even perfect number written in base 16.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 4, 7, 15, 22, 26, 31, 130, 151, 319, 550, 570, 804, 1063, 1105, 2422, 2485, 2803, 4984, 5425, 5802, 11124, 21560, 27625, 33012, 54022, 189209, 214858, 314446, 349567, 744055, 755344, 1743148, 3366729, 5249002, 6009145, 6491237, 7600614, 8145664, 9289166

Offset: 1

Juri-Stepan Gerasimov, Jun 11 2009

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Ivan Panchenko)

Cf. A000396, A135650, A160962, A161323.

a(n) = floor((A000043(n)-1)/4). - Gerson Washiski Barbosa, May 29 2011

Definition changed (inserting the word "even"), a(31) corrected and a(41)-a(45) added by Ivan Panchenko, Aug 04 2018

A211979 Numbers n formed by p 1's followed by p - 1 0's, where p is prime(n).

Original entry on oeis.org

110, 11100, 111110000, 1111111000000, 111111111110000000000, 1111111111111000000000000, 111111111111111110000000000000000, 1111111111111111111000000000000000000, 111111111111111111111110000000000000000000000

Offset: 1

Omar E. Pol, Dec 12 2012

			For n = 3, the third prime is 5, so a(3) = 111110000 (five 1's followed by four 0's).

Binary representation of A060286.
a(n) has A076274(n) digits.
Cf. A007088, A076274, A109241, A135650.

(* Technically this is in base 10 *) Table[10^(Prime[n] - 1)((10^Prime[n] - 1)/9), {n, 20}] (* Alonso del Arte, Dec 12 2012 *)
FromDigits[Join[PadRight[{},#,1],PadRight[{},#-1,0]]]&/@ Prime[ Range[ 10]] (* Harvey P. Dale, Aug 30 2015 *)

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A090748 Numbers k such that 2^(k+1) - 1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A133028 Even perfect numbers divided by 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A133033 Number of proper divisors of n-th even perfect number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A160962 Perfect numbers written in base 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A161323 Perfect numbers written in base 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A135651 Even superperfect numbers written in base 2.

Views

Author

Keywords

Comments

Examples