

A349606


Dirichlet convolution of the binary digital sum function (A000120) with itself.


1



1, 2, 4, 3, 4, 8, 6, 4, 8, 8, 6, 12, 6, 12, 16, 5, 4, 16, 6, 12, 18, 12, 8, 16, 10, 12, 16, 18, 8, 32, 10, 6, 16, 8, 18, 24, 6, 12, 20, 16, 6, 36, 8, 18, 32, 16, 10, 20, 15, 20, 16, 18, 8, 32, 22, 24, 20, 16, 10, 48, 10, 20, 36, 7, 16, 32, 6, 12, 22, 36, 8, 32
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OFFSET

1,2


LINKS



FORMULA

a(n) = 2 * A000120(n) if and only if n is a prime.
a(2^n) = n + 1.
a(n) == 1 (mod 2) if and only if n is a square of an odious number (A000069).
Sum_{k=1..n} a(k) ~ n * log(n)^3/(24 * log(2)^2) + O(n * log(n)^2) (Srichan, 2019).


MATHEMATICA

s[n_] := DigitCount[n, 2, 1]; a[n_] := DivisorSum[n, s[#] * s[n/#] &]; Array[a, 100]


PROG

(PARI) a(n) = sumdiv(n, d, hammingweight(d)*hammingweight(n/d)); \\ Michel Marcus, Nov 23 2021


CROSSREFS



KEYWORD

nonn,base


AUTHOR



STATUS

approved



