A349606 - OEIS

%I #10 Nov 28 2021 02:54:45

%S 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,

%T 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,

%U 32,22,24,20,16,10,48,10,20,36,7,16,32,6,12,22,36,8,32

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

%H Amiram Eldar, <a href="/A349606/b349606.txt">Table of n, a(n) for n = 1..10000</a>

%H Teerapat Srichan, <a href="https://doi.org/10.7546/nntdm.2019.25.1.122-127">Averages of the Dirichlet convolution of the binary digital sum</a>, Notes on Number Theory and Discrete Mathematics, Vol. 25, No. 1 (2019), pp. 122—127.

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

%F a(n) = 2 * A000120(n) if and only if n is a prime.

%F a(2^n) = n + 1.

%F a(n) == 1 (mod 2) if and only if n is a square of an odious number (A000069).

%F Sum_{k=1..n} a(k) ~ n * log(n)^3/(24 * log(2)^2) + O(n * log(n)^2) (Srichan, 2019).

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

%o (PARI) a(n) = sumdiv(n, d, hammingweight(d)*hammingweight(n/d)); \\ _Michel Marcus_, Nov 23 2021

%Y Cf. A000069, A000120.

%K nonn,base

%O 1,2

%A _Amiram Eldar_, Nov 23 2021