cp's OEIS Frontend

A291901 Numbers n such that the sum of sums of elements of subsets of divisors of n is a perfect number (A000396).

%I A291901 #26 Sep 22 2024 10:54:16
%S A291901 2,4,13,16,64,4096,65536,262144,3145341,932181397,1073741824,
%T A291901 1152921504606846976,309485009821345068724781056,
%U A291901 81129638414606681695789005144064,85070591730234615865843651857942052864,75603657215035519123837860069507929970384679
%N A291901 Numbers n such that the sum of sums of elements of subsets of divisors of n is a perfect number (A000396).
%C A291901 Numbers n such that A229335(n) is in the sequence of perfect numbers, A000396.
%C A291901 Corresponding values of perfect numbers: 6, 28, 28, 496, 8128, 33550336, 8589869056, 137438691328, 33550336, ...
%C A291901 All even superperfect numbers A061652(n) are terms in this sequence.
%C A291901 Primes q of the form 2^(p-2) * (2^p - 1) - 1 where p is a Mersenne exponent (A000043) are terms: 2, 13, ...
%H A291901 Max Alekseyev, <a href="/A291901/b291901.txt">Table of n, a(n) for n = 1..71</a>
%e A291901 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 A291901 sigma(16) * 2^(tau(16) - 1) = 31 * 16 = 496 (perfect number).
%p A291901 isA000396 := proc(n)
%p A291901     numtheory[sigma](n)=2*n ;
%p A291901     simplify(%) ;
%p A291901 end proc:
%p A291901 for n from 1 do
%p A291901     if isA000396(A229335(n)) then
%p A291901         print(n);
%p A291901     end if;
%p A291901 end do: # _R. J. Mathar_, Nov 10 2017
%t A291901 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 A291901 (Magma) [n: n in [1..10^6]  | SumOfDivisors(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1))) eq 2*(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1)))];
%Y A291901 Cf. A000043, A000396, A061652, A090748, A229335.
%K A291901 nonn
%O A291901 1,1
%A A291901 _Jaroslav Krizek_, Nov 02 2017
%E A291901 Terms a(10) onward added by _Max Alekseyev_, Sep 18 2024