A349606 Dirichlet convolution of the binary digital sum function (A000120) with itself.

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

Offset

1,2

Links

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

Teerapat Srichan, Averages of the Dirichlet convolution of the binary digital sum, Notes on Number Theory and Discrete Mathematics, Vol. 25, No. 1 (2019), pp. 122—127.

Formula

a(n) = Sum_{d|n} A000120(d) * A000120(n/d).

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

Cf. A000069, A000120.

Sequence in context: A335841 A133702 A328486 * A332224 A080001 A178938

Adjacent sequences: A349603 A349604 A349605 * A349607 A349608 A349609

Keyword

nonn,base

Author

Amiram Eldar, Nov 23 2021

Status

approved