A051027 -id:A051027 - cp's OEIS frontend

A336549 Numbers k, not powers of primes, such that A051027(k) = Product_{p^e|k} A051027(p^e). Here each p^e is the maximal power of prime p that divides k, and A051027(k) = sigma(sigma(k)).

6, 12, 14, 18, 20, 24, 26, 28, 36, 38, 44, 45, 48, 50, 54, 56, 62, 63, 68, 72, 74, 75, 76, 80, 86, 92, 96, 99, 100, 104, 108, 112, 116, 117, 122, 124, 126, 134, 144, 146, 147, 148, 150, 153, 158, 162, 171, 172, 175, 176, 180, 188, 192, 194, 196, 200, 206, 207, 208, 212, 218, 224, 225, 234, 236, 242, 244, 248, 252, 254

Offset: 1

Antti Karttunen, Jul 25 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences related to sigma(n)

Terms of A336547 without A000961.
Cf. A051027, A336546, A336548.
Subsequence of A336559. Cf also A336560.

isA336549(n) = ((n>1) && !isprimepower(n) && A336546(n)); \\ Needs also code from A336546.

A353806 a(n) = A353802(n) / gcd(A051027(n), A353802(n)), where A051027(n) = sigma(sigma(n)), and A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 5, 16, 1, 1, 1, 1, 1, 1, 1, 112, 1, 1, 49, 13, 45, 1, 1, 1, 7, 16, 1, 5, 1, 1, 1, 16, 1, 1, 1, 1, 7, 64, 1, 1, 112, 1, 49, 16, 1, 7, 1, 1, 1, 1, 9, 784, 1, 1, 5, 720, 1, 1, 1, 1, 1, 1, 5, 7, 1, 1, 1, 16, 1, 5, 117, 1, 7, 16, 1, 16, 45, 1, 147, 16, 7

Offset: 1

Antti Karttunen, May 08 2022

Numerator of fraction A353802(n) / A051027(n).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A051027, A353802, A353803, A353804, A353805 (denominators).
Cf. A336547 (positions of 1's), A336548 (positions of terms > 1), see also A353807.
Cf. also A353755, A353756.

A051027(n) = sigma(sigma(n));
A353806(n) = { my(f = factor(n), u=prod(k=1, #f~, A051027(f[k, 1]^f[k, 2]))); (u / gcd(A051027(n), u)); };

a(n) = A353802(n) / A353804(n) = A353802(n) / gcd(A051027(n), A353802(n)).

A336560 Numbers k at which points A336456(k) appears multiplicative, but A051027(k) does not.

Original entry on oeis.org

15, 39, 51, 60, 78, 87, 95, 111, 123, 143, 159, 183, 204, 215, 219, 222, 231, 240, 247, 267, 291, 303, 312, 323, 327, 330, 335, 339, 348, 366, 380, 399, 407, 411, 438, 444, 447, 455, 471, 494, 506, 519, 543, 559, 579, 582, 591, 624, 636, 654, 671, 687, 695, 699, 703, 714, 723, 731, 732, 767, 771, 779, 798, 803, 807

Offset: 1

Antti Karttunen, Jul 25 2020

nonn

Numbers in A336557 but not in A336547.

Note that if A051027(k) = Product_{p^e|k} A051027(p^e) then also A336456(n) = Product_{p^e|n} A336456(p^e), because A336456(n) = A335915(A051027(n)) and A335915 is fully multiplicative, thus A336547 is a subsequence of A336557.

Antti Karttunen, Table of n, a(n) for n = 1..25000

Cf. A051027, A335915, A336456.
Setwise difference of A336557 and A336547. Equally, setwise difference of A336559 and A336549. Subsequence of A336548.
Cf. also A336561.

is_fun_mult_on_n(fun,n) = { my(f=factor(n)); prod(k=1,#f~,fun(f[k,1]^f[k,2]))==fun(n); };
A051027(n) = sigma(sigma(n));
A000265(n) = (n>>valuation(n,2));
A335915(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265((f[k,1]^2)-1)^f[k,2]))); };
A336456(n) = A335915(A051027(n));
A336546(n) = is_fun_mult_on_n(A051027,n);
A336556(n) = is_fun_mult_on_n(A336456,n);
isA336560(n) = (A336546(n)<A336556(n));

A353805 a(n) = A051027(n) / gcd(A051027(n), A353802(n)), where A051027(n) = sigma(sigma(n)), and A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 13, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 31, 10, 31, 1, 1, 1, 5, 13, 1, 3, 1, 1, 1, 13, 1, 1, 1, 1, 5, 57, 1, 1, 65, 1, 31, 13, 1, 5, 1, 1, 1, 1, 7, 403, 1, 1, 3, 403, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 13, 1, 3, 70, 1, 5, 13, 1, 13, 31, 1, 85, 13, 5, 1, 1, 13

Offset: 1

Antti Karttunen, May 08 2022

Denominator of fraction A353802(n) / A051027(n).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A051027, A353802, A353803, A353804, A353806 (numerators).
Positions of 1's is given by the union of A336547 and A353807.
Cf. also A353755, A353756.

A051027(n) = sigma(sigma(n));
A353805(n) = { my(f = factor(n)); (A051027(n) / gcd(A051027(n), prod(k=1, #f~, A051027(f[k, 1]^f[k, 2])))); };

a(n) = A051027(n) / A353804(n).

A098219 a(n) = floor(sigma(sigma(n))/n) = floor(A051027(n)/n).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 1, 3, 2, 4, 1, 4, 4, 2, 2, 3, 2, 4, 3, 4, 2, 7, 1, 3, 3, 4, 2, 6, 2, 3, 3, 3, 3, 3, 1, 4, 3, 5, 2, 6, 1, 5, 3, 4, 2, 4, 1, 2, 3, 3, 2, 6, 3, 6, 3, 4, 2, 8, 1, 4, 3, 2, 3, 6, 1, 4, 3, 5, 2, 4, 1, 3, 2, 4, 3, 6, 2, 4, 1, 3, 2, 6, 3, 3, 4, 6, 2, 6, 2, 5, 2, 4, 3, 7, 1, 2, 3, 2, 2, 5, 2, 5, 4

Offset: 1

Labos Elemer, Oct 25 2004

nonn

			For n=10, a(10) = floor(sigma(sigma(10))/10) = floor(sigma(18)/10) = floor(39/10) = 3.

Indranil Ghosh, Table of n, a(n) for n = 1..25000

Cf. A000203, A051027, A008333, A098220, A098221, A098222.

Table[Floor[Nest[DivisorSigma[1, #] &, n, 2]/n], {n, 120}] (* Michael De Vlieger, Mar 02 2017 *)

a(n) = sigma(sigma(n))\n; \\ Michel Marcus, Oct 07 2019

A336561 Numbers k at which point A336459(k) appears multiplicative, but A051027(k) does not.

Original entry on oeis.org

506, 1819, 2024, 2714, 3674, 3818, 4554, 5088, 5750, 5786, 6026, 6762, 6842, 7215, 7276, 9487, 9523, 10442, 11895, 12397, 12650, 13178, 13303, 14235, 14696, 15272, 15962, 16346, 16371, 18216, 18458, 19274, 19514, 19690, 19706, 20179, 20378, 21079, 21255, 21626, 22066, 22586, 22682, 23000, 23144, 23322, 24104, 24246

Offset: 1

Antti Karttunen, Jul 25 2020

nonn

			506 = 2*11*23 is a term as A336459(2)*A336459(11)*A336459(23) = 1*7*5 = 35 = A336459(506), while A051027(2)*A051027(11)*A051027(23) = 4*28*60 = 6720 <> A051027(506) = 2520. Note that 2520 = 2^3 * 3^2 * 5 * 7, thus A065330(2520) = 5*7 = 35.

Antti Karttunen, Table of n, a(n) for n = 1..11481
Index entries for sequences related to sigma(n)

Cf. A051027, A065330, A336459.
Cf. also A336549.
Subsequence of A336548, and probably also of A336560.

is_fun_mult_on_n(fun,n) = { my(f=factor(n)); prod(k=1,#f~,fun(f[k,1]^f[k,2]))==fun(n); };
A051027(n) = sigma(sigma(n));
A336546(n) = is_fun_mult_on_n(A051027,n);
A065330(n) = (n>>valuation(n, 2)/3^valuation(n, 3));
A336459(n) = A065330(A051027(n));
isA336561(n) = (A336546(n)A336459,n));

A309530 Number of power-of-two-divisors of sum of divisors of sum of divisors of powers of two: a(n) = A001511(A051027(A000079(n))).

Original entry on oeis.org

1, 3, 4, 4, 6, 4, 8, 5, 5, 10, 5, 6, 14, 12, 12, 6, 18, 10, 20, 11, 9, 9, 6, 8, 8, 18, 6, 15, 7, 16, 32, 7, 11, 22, 17, 14, 7, 24, 22, 13, 5, 13, 11, 12, 20, 10, 7, 11, 9, 16, 33, 22, 6, 10, 15, 17, 28, 12, 6, 20, 62, 36, 12, 9, 24, 16, 5, 26, 12, 26, 10, 18, 6, 12, 16, 28, 19, 26

Offset: 0

Juri-Stepan Gerasimov, Aug 06 2019

nonn

			a(0) = A001511(A051027(A000079(0))) = A001511(A051027(A000079(2^0))) = A001511(A051027(1)) = A001511(1) = 1.

Juri-Stepan Gerasimov, x = A001511(A051027(A000079(x))), SeqFan list, Aug 19 2019.

Cf. A000043 (numbers m such that m - 1 divides a(m - 1) - 2), A000079, A001511, A051027, A090748.

[Valuation(2*SumOfDivisors(SumOfDivisors(2^n)),2): n in [0..89]];

a(n) = valuation(2*sigma(sigma(2^n)), 2); \\ Michel Marcus, Aug 06 2019

from sympy import divisor_sigma
def A309530(n): return ((m:=int(divisor_sigma((1<Chai Wah Wu, Jul 13 2022

a(n) = A001511(A051027(A000079(n))).

A162967 Values taken by the sigma(sigma(n)) function A051027, with repetition, sorted into ascending order.

Original entry on oeis.org

1, 4, 7, 8, 12, 14, 15, 24, 24, 28, 28, 32, 32, 39, 39, 42, 56, 56, 60, 60, 60, 60, 63, 63, 72, 80, 84, 90, 91, 96, 96, 96, 96, 104, 112, 114, 120, 120, 120, 120, 124, 124, 124, 126, 128, 128, 133, 160, 168, 168, 168, 168, 171, 171, 186, 186, 195, 195, 195, 195, 195

Offset: 1

Jaroslav Krizek, Jul 19 2009

nonn

Removal of duplicates generates A070286. - R. J. Mathar, Jul 21 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A007609, A002191. - R. J. Mathar, Jul 21 2009

Cf. A051027, A070286.

f(k) = {my(v = invsigma(k), c = 0); for(i = 1, #v, c += invsigmaNum(v[i])); c;} \\ using Max Alekseyev's invphi.gp
list(lim) = {my(m); for(k = 1, lim, m = f(k); for(i = 1, m, print1(k, ", ")));} \\ Amiram Eldar, Dec 26 2024

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

A019278 Numbers j such that sigma(sigma(j)) = k*j for some k.

Original entry on oeis.org

1, 2, 4, 8, 15, 16, 21, 24, 42, 60, 64, 84, 160, 168, 240, 336, 480, 504, 512, 960, 1023, 1344, 1536, 4092, 4096, 10752, 13824, 16368, 29127, 32256, 32736, 47360, 57120, 58254, 61440, 65472, 65536, 86016, 116508, 217728, 262144, 331520, 343976, 466032, 550095

Offset: 1

N. J. A. Sloane

nonn

Let sigma^m (j) be the result of applying the sum-of-divisors function (A000203) m times to j; call j (m,k)-perfect if sigma^m (j) = k*j; then this is the sequence of (2,k)-perfect numbers.
From Michel Marcus, May 14 2016: (Start)
For these numbers, the quotient k = sigma(sigma(j))/j is an integer (see A098223). Then also k = (sigma(s)/s)*(sigma(j)/j) with s = sigma(j). That is, k = abundancy(s)*abundancy(j).
So looking at the abundancy of these terms may be interesting. Indeed we see that 459818240 and 51001180160 are actually 3-perfect numbers (A005820), and the reason they are here is that they are coprime to 3. So their sums of divisors are 4-perfect numbers (A027687), yielding q=12.
In a similar way, we can see that the 5-perfect numbers (A046060) that are coprime to 5 will be terms of this sequence with q=30. There are 20 such numbers, the smallest being 13188979363639752997731839211623940096. (End)
From Michel Marcus, May 15 2016: (Start)
It is also interesting to note that for a(2)=8, s=sigma(8)=15 is also a term. This happens to be the case for chains of several terms in a row:
8, 15, 24, 60, 168, 480 with k = 3,4,7,8,9,10;
512, 1023, 1536, 4092, 10752, 32736 with k = 3,4,7,8,9,10;
29127, 47360, 116508, 331520, 932064, 2983680 with k = 4,7,8,9,13,14;
1556480, 3932040, 14008320 with k = 9,13,14;
106151936, 251650560, 955367424 with k = 9,13,14;
312792480, 1505806848 with k = 19,20;
6604416000, 30834059256 with k = 19,20;
9623577600, 46566269568 with k = 19,20.
When j is a term, we can test if s=sigma(j) is also a term; this way we get 6 more terms: 572941926400, 845734196736, 1422976331052, 4010593484800, 11383810648416, 36095341363200.
And the corresponding chains are:
173238912000, 845734196736 with k = 19,20;
355744082763, 572941926400, 1422976331052, 4010593484800, 11383810648416, 36095341363200 with k = 4,7,8,9,13,14. (End)
From Altug Alkan, May 17 2016: (Start)
Here are additional chains for the above list:
57120, 217728 with k = 13,14;
343976, 710400 with k = 7,8;
1980342, 5621760 with k = 10,14;
4404480, 14913024 with k = 11,12;
238608384, 775898880 with k = 11,12. (End)
Currently, the coefficient pairs are [1, 1], [3, 4], [4, 7], [7, 8], [8, 9], [9, 10], [9, 13], [10, 14], [11, 12], [13, 14], [16, 17], [16, 21], [17, 18], [19, 20], [23, 24], [25, 26], [25, 31], [27, 28], [29, 30], [31, 32], [32, 33], [37, 38]. It is interesting to note that for some of them, the pair (s,t) also satisfies t=sigma(s). - Michel Marcus, Jul 03 2016; Sep 06 2016
Using these empirical pairs of coefficients in conjunction with the first comment allows us to determine whether some term is the sum of divisors of another yet unknown smaller term. - Michel Marcus, Jul 04 2016
For m in A090748 = A000043 - 1 and c in A205597 (= odd a(n)), c*2^m is in the sequence, unless 2^(m+1)-1 | sigma(c). Indeed, from sigma(x*y) = sigma(x)*sigma(y) for gcd(x,y) = 1, we get sigma(sigma(c*2^m)) = sigma(sigma(c))*2^(m+1), so c*2^m is in the sequence if sigma(sigma(c))/c = k/2 (where k can't be odd: A330598 has no odd c). - M. F. Hasler, Jan 06 2020

Giovanni Resta, Table of n, a(n) for n = 1..145 (first 130 terms from Jud McCranie)
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Michel Marcus, List of terms, grouped by quotient, not exhaustive.

Cf. A098219, A098220, A098221, A098222, A098223, A008333, A051027, A019276.
For sigma see A000203 and A007691.
Cf. A205597 (odd terms), A323653 (those terms that are in A007691, i.e., for which sigma(n)/n is also an integer), A330598 (half-integer ratio).

[m: m in [1..560000]| IsIntegral(DivisorSigma(1,DivisorSigma(1,m))/m)]; // Marius A. Burtea, Nov 16 2019

Select[Range[100000], Mod[DivisorSigma[1, DivisorSigma[1, #]], #] == 0 &] (* Carl Najafi, Aug 22 2011 *)

is_A019278(n)=sigma(sigma(n))%n==0 \\ M. F. Hasler, Jul 02 2016

from sympy.ntheory import divisor_sigma as D
print([i for i in range(1, 10000) if D(D(i, 1), 1)%i==0]) # Indranil Ghosh, Mar 17 2017

Simpler definition from M. F. Hasler, Jul 02 2016

cp's OEIS Frontend

A336549 Numbers k, not powers of primes, such that A051027(k) = Product_{p^e|k} A051027(p^e). Here each p^e is the maximal power of prime p that divides k, and A051027(k) = sigma(sigma(k)).

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A353806 a(n) = A353802(n) / gcd(A051027(n), A353802(n)), where A051027(n) = sigma(sigma(n)), and A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).

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A336560 Numbers k at which points A336456(k) appears multiplicative, but A051027(k) does not.

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A353805 a(n) = A051027(n) / gcd(A051027(n), A353802(n)), where A051027(n) = sigma(sigma(n)), and A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).

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A098219 a(n) = floor(sigma(sigma(n))/n) = floor(A051027(n)/n).

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Mathematica

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A336561 Numbers k at which point A336459(k) appears multiplicative, but A051027(k) does not.

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A309530 Number of power-of-two-divisors of sum of divisors of sum of divisors of powers of two: a(n) = A001511(A051027(A000079(n))).

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Magma

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A162967 Values taken by the sigma(sigma(n)) function A051027, with repetition, sorted into ascending order.

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

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