A072868 -id:A072868 - cp's OEIS frontend

A075398 Duplicate of A072868.

4, 8, 32, 128, 8192, 131072, 524288, 2147483648, 2305843009213693952, 618970019642690137449562112, 162259276829213363391578010288128, 170141183460469231731687303715884105728

Offset: 1

Zak Seidov, Oct 11 2002

dead

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

A019279 Superperfect numbers: numbers k such that sigma(sigma(k)) = 2*k where sigma is the sum-of-divisors function (A000203).

Original entry on oeis.org

2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864

Offset: 1

N. J. A. Sloane

Let sigma_m(n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives (2,2)-perfect numbers.
Even values of these are 2^(p-1) where 2^p-1 is a Mersenne prime (A000043 and A000668). No odd superperfect numbers are known. Hunsucker and Pomerance checked that there are no odd ones below 7 * 10^24. - Jud McCranie, Jun 01 2000
The number of divisors of a(n) is equal to A000043(n), if there are no odd superperfect numbers. - Omar E. Pol, Feb 29 2008
The sum of divisors of a(n) is the n-th Mersenne prime A000668(n), provided that there are no odd superperfect numbers. - Omar E. Pol, Mar 11 2008
Largest proper divisor of A072868(n) if there are no odd superperfect numbers. - Omar E. Pol, Apr 25 2008
This sequence is a divisibility sequence if there are no odd superperfect numbers. - Charles R Greathouse IV, Mar 14 2012
For n>1, sigma(sigma(a(n))) + phi(phi(a(n))) = (9/4)*a(n). - Farideh Firoozbakht, Mar 02 2015
The term "super perfect number" was coined by Suryanarayana (1969). He and Kanold (1969) gave the general form of even superperfect numbers. - Amiram Eldar, Mar 08 2021

			sigma(sigma(4))=2*4, so 4 is in the sequence.

Dieter Bode, Über eine Verallgemeinerung der vollkommenen Zahlen, Dissertation, Braunschweig, 1971.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B9, pp. 99-100.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, pp. 110-111.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, pp. 38-42.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

P. Bundschuh, Aufgabe 601, Elem. Math., Vol. 24 (1969), p. 69; alternative link.
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
G. G. Dandapat, J. L. Hunsucker, and Carl Pomerance, Some new results on odd perfect numbers, Pacific J. Math. Volume 57, Number 2 (1975), 359-364.
A. Hoque and H. Kalita, Generalized perfect numbers connected with arithmetic functions, Math. Sci. Lett. 3, No. 3, 249-253 (2014).
J. L. Hunsucker and Carl Pomerance, There are no odd superperfect number less than 7*10^24, Indian J. Math., Vol. 17 (1975), pp. 107-120.
H.-J. Kanold, Über "Super perfect numbers", Elem. Math., Vol. 24 (1969), pp. 61-62; alternative link.
Graham Lord, Even Perfect and Superperfect Numbers, Elem. Math., Vol. 30 (1975), pp. 87-88.
H. G. Niederreiter, Solution of Aufgabe 601, Elem. Math., Vol. 25 (1970), pp. 66-67; alternative link.
Paul Shubhankar, Ten Problems of Number Theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013.
D. Suryanarayana, Super perfect numbers, Elem. Math., Vol. 24 (1969), pp. 16-17; alternative link.
D. Suryanarayana, There is no superperfect number of the form p^(2*alpha), Elem. Math., Vol. 28 (1973), pp. 148-150; alternative link.
László Tóth, The alternating sum-of-divisors function, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siófok, Hungary.
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
Eric Weisstein's World of Mathematics, Superperfect Number.
Wikipedia, Superperfect number.
Tomohiro Yamada, On finiteness of odd superperfect numbers, Journal de Théorie des Nombres de Bordeaux, Vol. 32, No. 1 (2020), pp. 259-274.

Cf. A019280, A000203, A000396, A000668, A000043, A034897, A061652, A032742, A072868.

sigma = DivisorSigma[1, #]&;
For[n = 2, True, n++, If[sigma[sigma[n]] == 2 n, Print[n]]] (* Jean-François Alcover, Sep 11 2018 *)

is(n)=sigma(sigma(n))==2*n \\ Charles R Greathouse IV, Nov 20 2012

from itertools import count, islice
def A019279_gen(): # generator of terms
    return (n for n in count(1) if divisor_sigma(divisor_sigma(n)) == 2*n)
A019279_list = list(islice(A019279_gen(),6)) # Chai Wah Wu, Feb 18 2022

a(n) = (1 + A000668(n))/2, if there are no odd superperfect numbers. - Omar E. Pol, Mar 11 2008
Also, if there are no odd superperfect numbers then a(n) = 2^A000043(n)/2 = A072868(n)/2 = A032742(A072868(n)). - Omar E. Pol, Apr 25 2008
a(n) = 2^A090748(n), if there are no odd superperfect numbers. - Ivan N. Ianakiev, Sep 04 2013

a(8)-a(9) from Jud McCranie, Jun 01 2000

Corrected by Michel Marcus, Oct 28 2017

A061652 Even superperfect numbers: 2^(p-1) where 2^p-1 is a Mersenne prime (A000668).

Original entry on oeis.org

2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864

Offset: 1

Jason Earls, Jun 16 2001

It is conjectured that there are no odd superperfect numbers, in which case this coincides with A019279.
The number of divisors of a(n) is equal to A000043(n). - Omar E. Pol, Feb 29 2008
The sum of divisors of a(n) is equal to A000668(n), the n-th Mersenne prime. - Omar E. Pol, Mar 11 2008
Largest proper divisor of A072868(n). - Omar E. Pol, Apr 25 2008
Indices of hexagonal numbers (A000384) that are also even perfect numbers. [Omar E. Pol, Aug 26 2008]
Except for the first perfect number 6, this sequence is the greatest common divisor of a perfect number (A000396) and its arithmetic derivative (A003415). - Giorgio Balzarotti, Apr 21 2011
If n is in the sequence then n is a solution to the equation phi(sigma(x)) = 2x-2. It seems that there is no other solution to this equation. - Jahangeer Kholdi, Sep 09 2014
The sum of sums of elements of subsets of divisors of a(n), i.e. A229335(a(n)), is a perfect number (A000396). - Jaroslav Krizek, Nov 02 2017

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

Vincenzo Librandi, Table of n, a(n) for n = 1..18
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Eric Weisstein's World of Mathematics, Superperfect Number
Index to divisibility sequences

Cf. A000043, A000384, A000396, A000668, A019279, A032742, A072868.

2^(Select[Range[512], PrimeQ[2^# - 1] &] - 1) (* Alonso del Arte, Apr 22 2011 *)
2^(MersennePrimeExponent[Range[15]]-1) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 20 2021 *)

forprime(p=2,1e3,if(ispseudoprime(2^p-1),print1(2^(p-1)", "))) \\ Charles R Greathouse IV, Mar 14 2012

a(n) = 2^(A090748(n)). - Lekraj Beedassy, Dec 07 2007
a(n) = (1 + A000668(n))/2. - Omar E. Pol, Mar 11 2008
a(n) = 2^A000043(n)/2 = A072868(n)/2 = A032742(A072868(n)). - Omar E. Pol, Apr 25 2008

A139306 Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).

Original entry on oeis.org

8, 32, 512, 8192, 33554432, 8589934592, 137438953472, 2305843009213693952, 2658455991569831745807614120560689152, 191561942608236107294793378393788647952342390272950272

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Sum of n-th even perfect number and n-th even superperfect number.

Also, sum of n-th perfect number and n-th superperfect number, if there are no odd perfect and odd superperfect numbers, then the n-th perfect number is the difference between a(n) and the n-th superperfect number (see A135652, A135653, A135654 and A135655).

			a(5) = 33554432 because A000043(5) = 13 and 2^(2*13 - 1) = 2^25 = 33554432.
Also, if there are no odd perfect and odd superperfect numbers then we can write a(5) = A000396(5) + A019279(5) = A000396(5) + A061652(5) = 33554432.

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

Cf. A000079, A000396, A019279, A061645, A061652, A133033, A135652, A135653, A135654, A135655, A139286, A139294, A139307.

Cf. A000043, A000668, A072868.

2^(2 * MersennePrimeExponent[Range[10]] - 1) (* Amiram Eldar, Oct 17 2024 *)

a(n) = 2^(2*A000043(n) - 1). Also, a(n) = 2^A133033(n), if there are no odd perfect numbers. Also, a(n) = A000396(n) + A019279(n), if there are no odd perfect and odd superperfect numbers. Also, a(n) = A000396(n) + A061652(n), if there are no odd perfect numbers, then we can write: perfect number A000396(n) = a(n) - A061652(n).

a(n) = A061652(n)*(A000668(n)+1) = A061652(n)*A072868(n). - Omar E. Pol, Apr 13 2008

A072869 a(n) = sigma(sigma(n)-n), where sigma = A000203, sum of the divisors of n.

Original entry on oeis.org

1, 1, 4, 1, 12, 1, 8, 7, 15, 1, 31, 1, 18, 13, 24, 1, 32, 1, 36, 12, 24, 1, 91, 12, 31, 14, 56, 1, 96, 1, 32, 24, 42, 14, 72, 1, 36, 18, 93, 1, 120, 1, 90, 48, 42, 1, 140, 15, 44, 32, 72, 1, 144, 18, 127, 24, 63, 1, 280, 1, 54, 42, 104, 20, 168, 1, 90, 40, 114, 1, 168, 1, 90, 57

Offset: 2

Benoit Cloitre, Jul 27 2002

Antti Karttunen, Table of n, a(n) for n = 2..16384
Index entries for sequences related to sums of divisors

Cf. A000203 (sigma(n)), A001065 (sigma(n) - n), A051027.

Cf. A072868 (fixed points).

Table[DivisorSigma[1,DivisorSigma[1,n]-n],{n,2,80}] (* Harvey P. Dale, Sep 23 2013 *)

A072869(n) = sigma(sigma(n)-n); \\ Antti Karttunen, Oct 30 2017

a(n) = A000203(A001065(n)). - Michel Marcus, Nov 26 2013
For p prime, a(p) = 1. - Michel Marcus, Nov 19 2014
a(k) = k if k is one greater than a Mersenne prime. - Jarrod G. Sage, Dec 07 2024

A075408 Perfect powers pp such that pp+1 is prime.

Original entry on oeis.org

1, 4, 16, 36, 100, 196, 256, 400, 576, 676, 1296, 1600, 2916, 3136, 4356, 5476, 7056, 8100, 8836, 12100, 13456, 14400, 15376, 15876, 16900, 17956, 21316, 22500, 24336, 25600, 28900, 30976, 32400, 33856, 41616, 42436, 44100, 50176, 52900, 55696

Offset: 1

Zak Seidov, Oct 11 2002

Of 1110 pp's < 10^6, 112 are such that pp+1 is prime and only seven are such that pp-1 is prime (see Mersenne primes (A000668)).

			pp=324900 is OK because pp=570^2 and pp+1=324901 (prime).

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A001597 (perfect powers), A072868 (pp-1 is prime).

Cf. A002496.

pp = Join[ Select[ Range[56000], Apply[GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 & ]]; Select[pp, PrimeQ[ # + 1] & ]

from sympy import isprime
print([n**2 for n in range(1,240) if isprime(n**2 + 1)]) # Karl-Heinz Hofmann, Feb 02 2023

a(n) = A002496(n) - 1.

Edited by Robert G. Wilson v, Oct 14 2002

Edited by N. J. A. Sloane, Dec 17 2009 at the suggestion of Rick Shepherd

A139096 Infraperfect numbers: a(n) = 2^(2*p - 1) - 2^p, where p is A000043(n).

Original entry on oeis.org

4, 24, 480, 8064, 33546240, 8589803520, 137438429184, 2305843007066210304, 2658455991569831743501771111346995200, 191561942608236107294793377774818628309652252823388160

Offset: 1

Omar E. Pol, Apr 22 2008

nonn

Difference between n-th even perfect number and n-th even superperfect number A061652(n). Difference between n-th ultraperfect number A139306(n) and n-th Mersenne prime A000668(n), minus 1. Also, difference between n-th perfect number A000396(n) and n-th superperfect number A019279(n), if there are no odd perfect and superperfect numbers.

			a(2) = 24 because A000043(2) = 3 then 2^(2*3 - 1) - 2^3 = 2^5 - 2^3 = 32 - 8 = 24.

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

Cf. A000043, A000396, A000668, A019279, A061652, A072868, A139306.

Map[2^(2*#-1) - 2^# &, MersennePrimeExponent[Range[10]]] (* Amiram Eldar, Oct 17 2024 *)

a(n) = 2^(2*A000043(n) - 1) - 2^A000043(n) = A139306(n) - 2^A000043(n) = A139306(n) - A000668(n) - 1 = A139306(n) - (A000668(n)+1) = A139306(n) - 2*A061652(n) = A139306(n) - A072868(n).

More terms from R. J. Mathar, Feb 05 2010

A066617 Composites of form prime+1 containing a record number of prime factors.

Original entry on oeis.org

4, 8, 24, 32, 128, 384, 1152, 3584, 5120, 6144, 8192, 73728, 131072, 524288, 5505024, 10616832, 14680064, 18874368, 109051904, 169869312, 654311424, 738197504, 2147483648, 21474836480, 51539607552, 824633720832, 3710851743744

Offset: 1

G. L. Honaker, Jr., Jan 13 2002

nonn

The sequence contains all numbers of the form (Mersenne Prime)+1 as a subset. - Hugo Pfoertner, Sep 10 2004

			a(19)=109051904=13*2^23: 24 prime factors, a(20)=169869312=3^4*2^21: 25 prime factors, a(21)=654311424=13*3*2^24: 26 prime factors. a(19)-1, a(20)-1 and a(21)-1 are primes.

Cf. Mersenne Primes + 1: A072868(n)=A000668(n)+1.

{A066617(a, b) = local(p, c, d); forprime(p=a,b,d=bigomega(p+1); if(d>c,c=d; print1(p+1,",")))} A066617(3,10^7)

More terms from Jason Earls, Jan 15 2002
More terms from Hugo Pfoertner, Sep 10 2004
a(24)-a(27) from Donovan Johnson, Dec 08 2009

A379724 a(n) = A000668(n) - 1.

Original entry on oeis.org

2, 6, 30, 126, 8190, 131070, 524286, 2147483646, 2305843009213693950, 618970019642690137449562110, 162259276829213363391578010288126, 170141183460469231731687303715884105726

Offset: 1

Mats Granvik and Antti Karttunen, Dec 31 2024

nonn

Cf. A000396, A000668, A023900, A072868, A379725.

nn = 12; A023900[n_] := DivisorSum[n, MoebiusMu[#] # &]; Table[A023900[PerfectNumber[n]], {n, 1, nn}]

a(n) = A023900(A000396(n)), provided no odd perfect numbers exist.

Conjecture; a(n) = A072868(n) - 2. - Omar E. Pol, Jan 16 2025

cp's OEIS Frontend

A075398 Duplicate of A072868.

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

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A019279 Superperfect numbers: numbers k such that sigma(sigma(k)) = 2*k where sigma is the sum-of-divisors function (A000203).

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A061652 Even superperfect numbers: 2^(p-1) where 2^p-1 is a Mersenne prime (A000668).

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A139306 Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).

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A072869 a(n) = sigma(sigma(n)-n), where sigma = A000203, sum of the divisors of n.

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A075408 Perfect powers pp such that pp+1 is prime.

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