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

Author

Jason Earls, Jun 16 2001

Keywords

• nonn
• nice

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

References

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

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

Crossrefs

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

Programs

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 *)

PARI

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

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