A000668 -id:A000668 - cp's OEIS frontend

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

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

A046528 Numbers that are a product of distinct Mersenne primes (elements of A000668).

Original entry on oeis.org

1, 3, 7, 21, 31, 93, 127, 217, 381, 651, 889, 2667, 3937, 8191, 11811, 24573, 27559, 57337, 82677, 131071, 172011, 253921, 393213, 524287, 761763, 917497, 1040257, 1572861, 1777447, 2752491, 3120771, 3670009, 4063201, 5332341, 7281799, 11010027, 12189603

Offset: 1

Labos Elemer

nonn

Or, numbers n such that the sum of the divisors of n is a power of 2, see A094502.
Or, numbers n such that the number of divisors of n and the sum of the divisors of n are both powers of 2.
n is a product of distinct Mersenne primes iff sigma(n) is a power of 2: see exercise in Sivaramakrishnan, or Shallit.
Sequence gives n > 1 such that sigma(n) = 2*phi(sigma(n)). - Benoit Cloitre, Feb 22 2002
The graph of this sequence shows a discontinuity at the 4097th number because there is a large relative gap between the 12th and 13th Mersenne primes, A000043. Other discontinuities can be predicted using A078426. - T. D. Noe, Oct 12 2006
Supersequence of A051281 (numbers n such that sigma(n) is a power of tau(n)). Conjecture: numbers n such that sigma(n) = tau(n)^(a/b), where a, b are integers >= 1. Example: sigma(93) = 128 = tau(93)^(7/2) = 4^(7/2). - Jaroslav Krizek, May 04 2013

			a(20) = 82677 = 3*7*31*127, whose sum of divisors is 131072 = 2^17;
a(27) = 1040257 = 127*8191, whose sum of divisors is 1048576 = 2^20.

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problem 264 pp. 188, Ellipses Paris 2004.
R. Sivaramakrishnan, Classical Theory of Arithmetic Functions. Dekker, 1989.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from T. D. Noe)
Kevin S. Brown, Sum of Divisors Equals a Power of 2.
C. D. H. Cooper, Problem E 2493, The American Mathematical Monthly, Vol. 81, No. 8 (1974), p. 902; W. J. Dodge, solution, ibid., Vol. 82, No. 8 (1975), pp. 855-856.
Jeffrey Shallit, Problem 1319, Diophantine Equation, sigma(n) = 2^m, Math. Magazine, 63 (1990), 129.
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000668, A000043, A056652 (product of Mersenne primes), A306204.

mersennes:= [seq(numtheory:-mersenne([i]),i=1..10)]:
sort(select(`<`,map(convert,combinat:-powerset(mersennes),`*`),numtheory:-mersenne([11]))); # Robert Israel, May 01 2016

{1}~Join~TakeWhile[Times @@@ Rest@ Subsets@ # // Sort, Function[k, k <= Last@ #]] &@ Select[2^Range[0, 31] - 1, PrimeQ] (* Michael De Vlieger, May 01 2016 *)

isok(n) = (n==1) || (ispower(sigma(n), , &r) && (r==2)); \\ Michel Marcus, Dec 10 2013

Sum_{n>=1} 1/a(n) = Product_{p in A000668} (1 + 1/p) = 1.5855588879... (A306204) - Amiram Eldar, Jan 06 2021

More terms from Benoit Cloitre, Feb 22 2002
Further terms from Jon Hart, Sep 22 2006
Entry revised by N. J. A. Sloane, Mar 20 2007
Three more terms from Michel Marcus, Dec 10 2013

A139256 Twice even perfect numbers. Also a(n) = M(n)*(M(n)+1), where M(n) is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

12, 56, 992, 16256, 67100672, 17179738112, 274877382656, 4611686016279904256, 5316911983139663489309385231907684352, 383123885216472214589586756168607276261994643096338432

Offset: 1

Omar E. Pol, Apr 22 2008

nonn

Also, twice perfect numbers, if there are no odd perfect numbers.
If there are no odd perfect numbers, essentially the same as A065125. - R. J. Mathar, May 23 2008
The sum of reciprocals of even divisors of a(n) equals 1. Proof: Let n = (2^m - 1)*2^m where 2^m - 1 is a Mersenne prime. The sum of reciprocals of even divisors of n is s1 + s2 where: s1 = 1/2 + 1/4 + ... + 1/2^m = (2^m - 1)/2^m and s2 = s1/(2^m - 1) => s1 + s2 = 1. - Michel Lagneau, Jul 17 2013

			a(3) = 992 because the third Mersenne prime A000668(3) is 31 and 31*(31+1) = 31*32 = 992.
a(3) = 992 because the sum of the divisors of the third perfect number is 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496 = 992. - _Omar E. Pol_, Dec 05 2016
From _Omar E. Pol_, Aug 13 2021: (Start)
Illustration of initial terms in which a(n) is represented as the sum of the divisors of the n-th even perfect number P(n).
-------------------------------------------------------------------------
  n  P(n) a(n)  Diagram:   1                                           2
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                      |    _|                                         | |
                 _ _ _|  _|                                           | |
  1    6   12   |_ _ _ _|                                             | |
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                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  2   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) equals the area (also the number of cells) in the n-th diagram.
For n = 3, P(3) = 496 and a(3) = 992, the diagram is too large to include here. To draw that diagram note that the lengths of the line segments of the smallest Dyck path are [248, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 248] and the lengths of the line segments of the largest Dyck path are [249, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 249]. For a definition of these numbers related to partitions into consecutive parts see A237591. (End)

Walter A. Kehowski, Power-spectral Numbers, ResearchGate (2024); also available at vixra.org.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000203, A000396, A000668, A001065, A065125, A139257, A237591, A237593, A245092.

DeleteCases[2 Map[(# (# + 1))/2 &, Select[2^Range[100] - 1, PrimeQ]], k_ /; OddQ@ k] (* Michael De Vlieger, Dec 05 2016, after Harvey P. Dale at A000396 *)

a(n) = A000668(n)*(A000668(n)+1).
a(n) = 2*A000396(n), if there are no odd perfect numbers.
a(n) = A000203(A000396(n)) = A001065(A000396(n)) + A000396(n), assuming there are no odd perfect numbers. - Omar E. Pol, Dec 04 2016

More terms from Omar E. Pol, Jun 07 2012

A124477 Numbers k such that 24k+7 is a Mersenne prime (A000668).

Original entry on oeis.org

0, 1, 5, 341, 5461, 21845, 89478485, 96076792050570581, 25790417485112089060398421, 6760803201217223474649083762005, 7089215977519551322153637654828504405

Offset: 1

Artur Jasinski, Dec 17 2006

nonn

Note that 2^m - 1 can be expressed as 24*k+7 whenever m is an odd integer >= 3. - Robert Israel, Jul 08 2014

Michel Marcus, Table of n, a(n) for n = 1..18

Cf. A000043, A000668, A139480.

seq((numtheory:-mersenne([i+1])-7)/24, i=1..20); # Robert Israel, Jul 08 2014

for(n=0, 1e20, k=0; if(ispseudoprime(24*n+7), while(2^k-1 < 24*n+7, k++); if(24*n+7==2^k-1, print1(n, ", ")))) \\ Felix Fröhlich, Jul 04 2014

lista(nn) = {vmps = readvec("b000043.txt"); if (nn== 0, nn = #vmps); for (i=1, nn, mpi = 2^vmps[i]-8; if ((mpi % 24) == 0, print1(mpi/24, ", ")););} \\ Michel Marcus, Jul 05 2014

a(n) = (2^A000043(n+1)-8)/24. - Jeppe Stig Nielsen, Sep 17 2020

a(11) corrected by Michel Marcus, Jul 05 2014

A335431 Numbers of the form q*(2^k), where q is one of the Mersenne primes (A000668) and k >= 0.

Original entry on oeis.org

3, 6, 7, 12, 14, 24, 28, 31, 48, 56, 62, 96, 112, 124, 127, 192, 224, 248, 254, 384, 448, 496, 508, 768, 896, 992, 1016, 1536, 1792, 1984, 2032, 3072, 3584, 3968, 4064, 6144, 7168, 7936, 8128, 8191, 12288, 14336, 15872, 16256, 16382, 24576, 28672, 31744, 32512, 32764, 49152, 57344, 63488, 65024, 65528, 98304, 114688, 126976, 130048, 131056, 131071

Offset: 1

Antti Karttunen, Jun 28 2020

nonn

Numbers of the form 2^k * ((2^p)-1), where p is one of the primes in A000043, and k >= 0.
Numbers k such that A000265(k) is in A000668.
Numbers k for which A331410(k) = 1.
Numbers k that themselves are not powers of two, but for which A335876(k) = k+A052126(k) is [a power of 2].
Conjecture: This sequence gives all fixed points of map n -> A332214(n) and its inverse n -> A332215(n). See also notes in A029747 and in A163511.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000043, A000396 (even terms form a subsequence), A000668 (primes present), A335882, A341622.
Row 1 of A335430.
Positions of 1's in A331410, in A364260, and in A364251 (characteristic function).
Subsequence of A054784.
Cf. also A029747, A163511, A173898, A332214, A332215, A334101, A335876.

qs = 2^MersennePrimeExponent[Range[6]] - 1; max = qs[[-1]]; Reap[Do[n = 2^k*q; If[n <= max, Sow[n]], {k, 0, Log2[max]}, {q, qs}]][[2, 1]] // Union (* Amiram Eldar, Feb 18 2021 *)

A000265(n) = (n>>valuation(n,2));
isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA335431(n) = isA000668(A000265(n));

A332214(a(n)) = A332215(a(n)) = a(n) for all n.

Sum_{n>=1} 1/a(n) = 2 * A173898 = 1.0329083578... - Amiram Eldar, Feb 18 2021

A139294 a(n) = 2^(2p - 1), where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

32, 8192, 2305843009213693952, 14474011154664524427946373126085988481658748083205070504932198000989141204992

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Next terms have 4932, 78913, 315652, 1292913986, and 1388255822130839283 decimal digits. - Jens Kruse Andersen, Jul 14 2014

Cf. A000079, A000668, A139286, A139295, A139296, A139297, A139298, A139299, A139300, A139301, A139302, A139303, A139304, A139305, A139306.

A000668 := Select[2^Range[500] - 1, PrimeQ]; Table[2^(2*A000668[[n]] - 1), {n, 1, 5}] (* G. C. Greubel, Oct 03 2017 *)
a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 3); Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

\p 100
print1("a(n): "); forprime(q=2, 7, p=2^q-1; if(isprime(p), print1(2^(2*p-1)", ")));
print1("\nNumber of digits in a(n): "); forprime(q=2, 127, p=2^q-1; if(isprime(p), print1(ceil((2*p-1)*log(2)/log(10))", "))) \\ Jens Kruse Andersen, Jul 14 2014

a(n) = 2^(2*A000668(n)-1).

A248932 Decimal expansion of 2^2203 - 1, the 16th Mersenne prime A000668(16).

Original entry on oeis.org

1, 4, 7, 5, 9, 7, 9, 9, 1, 5, 2, 1, 4, 1, 8, 0, 2, 3, 5, 0, 8, 4, 8, 9, 8, 6, 2, 2, 7, 3, 7, 3, 8, 1, 7, 3, 6, 3, 1, 2, 0, 6, 6, 1, 4, 5, 3, 3, 3, 1, 6, 9, 7, 7, 5, 1, 4, 7, 7, 7, 1, 2, 1, 6, 4, 7, 8, 5, 7, 0, 2, 9, 7, 8, 7, 8, 0, 7, 8, 9, 4, 9, 3, 7, 7, 4, 0, 7, 3, 3, 7, 0, 4, 9, 3, 8, 9, 2, 8, 9, 3, 8, 2, 7, 4

Offset: 664

Arkadiusz Wesolowski, Oct 17 2014

The 13th through the 17th Mersenne primes were found in 1952 by Raphael M. Robinson, using SWAC.

			14759799152141802350848986227373817363120661453331697751477712164785702...

Arkadiusz Wesolowski, Table of n, a(n) for n = 664..1327
D. H. Lehmer, Two New Mersenne Primes, Mathematics of Computation, vol. 7, No. 41 (1952), p. 72.
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20).

Magma
```
Reverse(Intseq(2^2203-1));
```

RealDigits[2^2203-1,10,120][[1]] (* Harvey P. Dale, Oct 09 2017 *)

PARI
```
eval(Vec(Str(2^2203-1)))
```

2^A000043(16) - 1.

A204063 Decimal expansion of 2^607 - 1, the 14th Mersenne prime A000668(14).

Original entry on oeis.org

5, 3, 1, 1, 3, 7, 9, 9, 2, 8, 1, 6, 7, 6, 7, 0, 9, 8, 6, 8, 9, 5, 8, 8, 2, 0, 6, 5, 5, 2, 4, 6, 8, 6, 2, 7, 3, 2, 9, 5, 9, 3, 1, 1, 7, 7, 2, 7, 0, 3, 1, 9, 2, 3, 1, 9, 9, 4, 4, 4, 1, 3, 8, 2, 0, 0, 4, 0, 3, 5, 5, 9, 8, 6, 0, 8, 5, 2, 2, 4, 2, 7, 3, 9, 1, 6, 2, 5, 0, 2, 2, 6, 5, 2, 2, 9, 2, 8, 5, 6, 6, 8, 8, 8, 9

Offset: 183

M. F. Hasler, Jan 09 2013

			2^607-1 = 531 * 10^180 +
137992816767098689588206552468627329593117727031923199444138 * 10^120 +
200403559860852242739162502265229285668889329486246501015346 * 10^60 +
579337652707239409519978766587351943831270835393219031728127.

M. F. Hasler, Table of n, a(n) for n = 183..365
D. H. Lehmer, Recent Discoveries of Large Primes, Mathematics of Computation, vol. 6, No. 37 (1952), p. 61.
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A248931 = A000668(15), A248932 = A000668(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20).

Reverse(Intseq(2^607-1)); // Arkadiusz Wesolowski, Oct 18 2014

RealDigits[2^607-1,10,120][[1]] (* Harvey P. Dale, Sep 23 2016 *)

digits(2^607-1) \\ Simplified by M. F. Hasler, Oct 19 2019

2^A000043(14)-1.

A248931 Decimal expansion of 2^1279 - 1, the 15th Mersenne prime A000668(15).

Original entry on oeis.org

1, 0, 4, 0, 7, 9, 3, 2, 1, 9, 4, 6, 6, 4, 3, 9, 9, 0, 8, 1, 9, 2, 5, 2, 4, 0, 3, 2, 7, 3, 6, 4, 0, 8, 5, 5, 3, 8, 6, 1, 5, 2, 6, 2, 2, 4, 7, 2, 6, 6, 7, 0, 4, 8, 0, 5, 3, 1, 9, 1, 1, 2, 3, 5, 0, 4, 0, 3, 6, 0, 8, 0, 5, 9, 6, 7, 3, 3, 6, 0, 2, 9, 8, 0, 1, 2, 2, 3, 9, 4, 4, 1, 7, 3, 2, 3, 2, 4, 1, 8, 4, 8, 4, 2, 4

Offset: 386

Arkadiusz Wesolowski, Oct 17 2014

The 13th through the 17th Mersenne primes were found in 1952 by Raphael M. Robinson, using SWAC.

			10407932194664399081925240327364085538615262247266704805319112350403608...

Arkadiusz Wesolowski, Table of n, a(n) for n = 386..771
D. H. Lehmer, A New Mersenne Prime, Mathematics of Computation, vol. 6, No. 39 (1952), p. 205.
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248932 = A000668(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20).

Magma
```
Reverse(Intseq(2^1279-1));
```

RealDigits[2^1279 - 1, 10, 100][[1]] (* G. C. Greubel, Oct 03 2017 *)

PARI
```
eval(Vec(Str(2^1279-1)))
```

Equals 2^A000043(15) - 1.

A248933 Decimal expansion of 2^2281 - 1, the 17th Mersenne prime A000668(17).

Original entry on oeis.org

4, 4, 6, 0, 8, 7, 5, 5, 7, 1, 8, 3, 7, 5, 8, 4, 2, 9, 5, 7, 1, 1, 5, 1, 7, 0, 6, 4, 0, 2, 1, 0, 1, 8, 0, 9, 8, 8, 6, 2, 0, 8, 6, 3, 2, 4, 1, 2, 8, 5, 9, 9, 0, 1, 1, 1, 1, 9, 9, 1, 2, 1, 9, 9, 6, 3, 4, 0, 4, 6, 8, 5, 7, 9, 2, 8, 2, 0, 4, 7, 3, 3, 6, 9, 1, 1, 2, 5, 4, 5, 2, 6, 9, 0, 0, 3, 9, 8, 9, 0, 2, 6, 1, 5, 3

Offset: 687

Arkadiusz Wesolowski, Oct 17 2014

The 13th through the 17th Mersenne primes were found in 1952 by Raphael M. Robinson, using SWAC.

The digits of this prime were published on page 167 of Nordisk Mathematisk Tidskrift 2 (1954).

			44608755718375842957115170640210180988620863241285990111199121996340468...

Arkadiusz Wesolowski, Table of n, a(n) for n = 687..1373
D. H. Lehmer, Two New Mersenne Primes, Mathematics of Computation, vol. 7, No. 41 (1952), p. 72.
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000668(16), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20).

Magma
```
Reverse(Intseq(2^2281-1));
```

RealDigits[2^2281 - 1, 10, 100][[1]] (* G. C. Greubel, Oct 03 2017 *)

PARI
```
eval(Vec(Str(2^2281-1)))
```

Equals 2^A000043(17) - 1.

cp's OEIS Frontend

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

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A046528 Numbers that are a product of distinct Mersenne primes (elements of A000668).

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A139256 Twice even perfect numbers. Also a(n) = M(n)*(M(n)+1), where M(n) is the n-th Mersenne prime A000668(n).

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A124477 Numbers k such that 24k+7 is a Mersenne prime (A000668).

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A335431 Numbers of the form q*(2^k), where q is one of the Mersenne primes (A000668) and k >= 0.

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A139294 a(n) = 2^(2p - 1), where p is the n-th Mersenne prime A000668(n).

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A248932 Decimal expansion of 2^2203 - 1, the 16th Mersenne prime A000668(16).

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