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A291901
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Numbers n such that the sum of sums of elements of subsets of divisors of n is a perfect number (A000396).
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0
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OFFSET
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1,1
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COMMENTS
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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, ...
a(10) > 10^7; conjecture: a(10) = 1073741824.
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, ...
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LINKS
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EXAMPLE
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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).
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MAPLE
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isA000396 := proc(n)
numtheory[sigma](n)=2*n ;
simplify(%) ;
end proc:
for n from 1 do
print(n);
end if;
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MATHEMATICA
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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 *)
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PROG
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(Magma) [n: n in [1..10^6] | SumOfDivisors(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1))) eq 2*(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1)))]
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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