A349380 Dirichlet convolution of A003415 (arithmetic derivative of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

0, 1, 1, 3, 1, 2, 1, 8, 4, 3, 1, 5, 1, 4, 3, 20, 1, 6, 1, 8, 4, 6, 1, 12, 7, 7, 14, 11, 1, 3, 1, 48, 6, 9, 5, 14, 1, 10, 7, 20, 1, 4, 1, 17, 8, 12, 1, 28, 10, 13, 9, 20, 1, 18, 7, 28, 10, 15, 1, 6, 1, 16, 11, 112, 8, 6, 1, 26, 12, 5, 1, 32, 1, 19, 11, 29, 8, 7, 1, 48, 46, 21, 1, 8, 10, 22, 15, 44, 1, 6, 9, 35, 16

Offset

1,4

Comments

Dirichlet convolution of A349394 with A349432.

Dirichlet convolution with A349136 gives A300251.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = Sum_{d|n} A003415(n/d) * A349134(d).

a(n) = Sum_{d|n} A349394(n/d) * A349432(d).

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003602(n) = (1+(n>>valuation(n, 2)))/2;
memoA349134 = Map();
A349134(n) = if(1==n, 1, my(v); if(mapisdefined(memoA349134, n, &v), v, v = -sumdiv(n, d, if(d<n, A003602(n/d)*A349134(d), 0)); mapput(memoA349134, n, v); (v)));
A349380(n) = sumdiv(n, d, A003415(d)*A349134(n/d));

Crossrefs

Cf. A003415, A003602, A349134, A349394, A349432.

Cf. also A300251, A347955, A349136, A349431.

Sequence in context: A351436 A226629 A349620 * A351425 A249580 A340147

Adjacent sequences: A349377 A349378 A349379 * A349381 A349382 A349383

Keyword

sign,look

Author

Antti Karttunen, Nov 21 2021

Status

approved