

A061652


Even superperfect numbers: 2^(p1) where 2^p1 is a Mersenne prime (A000668).


61



2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864
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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 nth 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)) = 2x2. 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 sumofdivisors function, Experimental Mathematics, 5 (1996), pp. 93100.
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 *)


PROG

(PARI) forprime(p=2, 1e3, if(ispseudoprime(2^p1), print1(2^(p1)", "))) \\ Charles R Greathouse IV, Mar 14 2012


CROSSREFS

Cf. A000043, A000384, A000396, A000668, A019279, A032742, A072868.
Sequence in context: A061286 A288756 A019279 * A278913 A306979 A162119
Adjacent sequences: A061649 A061650 A061651 * A061653 A061654 A061655


KEYWORD

nonn,nice


AUTHOR

Jason Earls, Jun 16 2001


STATUS

approved



