A061652 - OEIS

%I #51 Jun 20 2021 13:08:37

%S 2,4,16,64,4096,65536,262144,1073741824,1152921504606846976,

%T 309485009821345068724781056,81129638414606681695789005144064,

%U 85070591730234615865843651857942052864

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

%C It is conjectured that there are no odd superperfect numbers, in which case this coincides with A019279.

%C The number of divisors of a(n) is equal to A000043(n). - _Omar E. Pol_, Feb 29 2008

%C The sum of divisors of a(n) is equal to A000668(n), the n-th Mersenne prime. - _Omar E. Pol_, Mar 11 2008

%C Largest proper divisor of A072868(n). - _Omar E. Pol_, Apr 25 2008

%C Indices of hexagonal numbers (A000384) that are also even perfect numbers. [_Omar E. Pol_, Aug 26 2008]

%C 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

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A061652/b061652.txt">Table of n, a(n) for n = 1..18</a>

%H G. L. Cohen and H. J. J. te Riele, <a href="http://www.emis.de/journals/EM/expmath/volumes/5/5.html">Iterating the sum-of-divisors function</a>, Experimental Mathematics, 5 (1996), pp. 93-100.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SuperperfectNumber.html">Superperfect Number</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%F a(n) = 2^(A090748(n)). - _Lekraj Beedassy_, Dec 07 2007

%F a(n) = (1 + A000668(n))/2. - _Omar E. Pol_, Mar 11 2008

%F a(n) = 2^A000043(n)/2 = A072868(n)/2 = A032742(A072868(n)). - _Omar E. Pol_, Apr 25 2008

%t 2^(Select[Range[512], PrimeQ[2^# - 1] &] - 1) (* _Alonso del Arte_, Apr 22 2011 *)

%t 2^(MersennePrimeExponent[Range[15]]-1) (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jun 20 2021 *)

%o (PARI) forprime(p=2,1e3,if(ispseudoprime(2^p-1),print1(2^(p-1)", "))) \\ _Charles R Greathouse IV_, Mar 14 2012

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

%K nonn,nice

%O 1,1

%A _Jason Earls_, Jun 16 2001