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

1,1

Comments

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

Links

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

Formula

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A061286 A288756 A019279 * A278913 A306979 A358032

Adjacent sequences: A061649 A061650 A061651 * A061653 A061654 A061655

Keyword

nonn,nice

Author

Jason Earls, Jun 16 2001

Status

approved