A291901 - OEIS

%I #26 Sep 22 2024 10:54:16

%S 2,4,13,16,64,4096,65536,262144,3145341,932181397,1073741824,

%T 1152921504606846976,309485009821345068724781056,

%U 81129638414606681695789005144064,85070591730234615865843651857942052864,75603657215035519123837860069507929970384679

%N Numbers n such that the sum of sums of elements of subsets of divisors of n is a perfect number (A000396).

%C Numbers n such that A229335(n) is in the sequence of perfect numbers, A000396.

%C Corresponding values of perfect numbers: 6, 28, 28, 496, 8128, 33550336, 8589869056, 137438691328, 33550336, ...

%C All even superperfect numbers A061652(n) are terms in this sequence.

%C Primes q of the form 2^(p-2) * (2^p - 1) - 1 where p is a Mersenne exponent (A000043) are terms: 2, 13, ...

%H Max Alekseyev, <a href="/A291901/b291901.txt">Table of n, a(n) for n = 1..71</a>

%e Divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; sum of sums of elements of subsets = 1 + 2 + 4 + 3 + 5 + 6 + 7 = 28 (perfect number).

%e sigma(16) * 2^(tau(16) - 1) = 31 * 16 = 496 (perfect number).

%p isA000396 := proc(n)

%p     numtheory[sigma](n)=2*n ;

%p     simplify(%) ;

%p end proc:

%p for n from 1 do

%p     if isA000396(A229335(n)) then

%p         print(n);

%p     end if;

%p end do: # _R. J. Mathar_, Nov 10 2017

%t Select[Range[2^20], DivisorSigma[1, DivisorSigma[1, #] 2^(DivisorSigma[0, #] - 1)] == 2 (DivisorSigma[1, #] 2^(DivisorSigma[0, #] - 1)) &] (* _Michael De Vlieger_, Nov 02 2017 *)

%o (Magma) [n: n in [1..10^6]  | SumOfDivisors(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1))) eq 2*(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1)))];

%Y Cf. A000043, A000396, A061652, A090748, A229335.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Nov 02 2017

%E Terms a(10) onward added by _Max Alekseyev_, Sep 18 2024