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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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
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
Sequence in context: A335841 A133702 A328486 * A332224 A080001 A178938
KEYWORD
nonn,base
AUTHOR
Amiram Eldar, Nov 23 2021
STATUS
approved

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Last modified April 21 01:42 EDT 2024. Contains 371850 sequences. (Running on oeis4.)