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

2, 4, 13, 16, 64, 4096, 65536, 262144, 3145341, 932181397, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864, 75603657215035519123837860069507929970384679

Offset: 1

Author

Jaroslav Krizek, Nov 02 2017

Keywords

• nonn

Comments

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

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

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

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

Examples

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

Links

• Max Alekseyev, Table of n, a(n) for n = 1..71

Crossrefs

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

Programs

Magma

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

Maple

isA000396 := proc(n)
    numtheory[sigma](n)=2*n ;
    simplify(%) ;
end proc:
for n from 1 do
    if isA000396(A229335(n)) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2017

Mathematica

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 *)

Extensions

Terms a(10) onward added by Max Alekseyev, Sep 18 2024