A380845 The sum of divisors of n that have the same binary weight as n.

1, 3, 3, 7, 5, 9, 7, 15, 12, 15, 11, 21, 13, 21, 15, 31, 17, 36, 19, 35, 28, 33, 23, 45, 25, 39, 27, 49, 29, 45, 31, 63, 36, 51, 42, 84, 37, 57, 39, 75, 41, 84, 43, 77, 60, 69, 47, 93, 56, 75, 51, 91, 53, 81, 55, 105, 57, 87, 59, 105, 61, 93, 63, 127, 70, 108

Offset

1,2

Comments

The number of these divisors is A380844(n).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} d * [A000120(d) = A000120(n)], where [ ] is the Iverson bracket.

a(2^n) = 2^(n+1) - 1.

a(n) <= A000203(n) with equality if and only if n is a power of 2.

a(n) = a(A000265(n)) * (2^(A007814(n)+1)-1) = a(A000265(n)) * A038712(n), or equivalently, a(k*2^n) = a(k)*(2^(n+1)-1) for k odd and n >= 0.

In particular, since a(p) = p for an odd prime p, a(p*2^n) = p*(2^(n+1)-1) for an odd prime p and n >= 0.

a(A000396(n)) = A000668(n)^2, assuming that odd perfect numbers do no exist.

Example

a(6) = 9 because 6 = 110_2 has binary weight 2, 2 of its divisors, 3 = 11_2 and 6, have the same binary weight, and 3 + 6 = 9.

Mathematica

a[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, # &, DigitCount[#, 2, 1] == h &]]; Array[a, 100]

Prog

(PARI) a(n) = {my(h = hammingweight(n)); sumdiv(n, d, d * (hammingweight(d) == h)); }

Crossrefs

Cf. A000120, A000203, A000265, A000396, A000668, A007814, A038712, A380844.

Sequence in context: A098688 A129266 A129527 * A245550 A318461 A085379

Adjacent sequences: A380842 A380843 A380844 * A380846 A380847 A380848

Keyword

nonn,base,easy

Author

Amiram Eldar, Feb 05 2025

Status

approved