A288417 a(n) = Sum_{d|n} A000593(n/d).

1, 2, 5, 3, 7, 10, 9, 4, 18, 14, 13, 15, 15, 18, 35, 5, 19, 36, 21, 21, 45, 26, 25, 20, 38, 30, 58, 27, 31, 70, 33, 6, 65, 38, 63, 54, 39, 42, 75, 28, 43, 90, 45, 39, 126, 50, 49, 25, 66, 76, 95, 45, 55, 116, 91, 36, 105, 62, 61, 105, 63, 66, 162, 7, 105, 130, 69

Offset: 1

Author

Seiichi Manyama, Jun 09 2017

Keywords

• nonn
• mult

Comments

Multiplicative because this sequence is the Dirichlet convolution of A000012 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000012.

Sum_{d|n} d^k*A000593(n/d): this sequence (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), A288420 (k=4).

Programs

Mathematica

f[p_, e_] := Sum[(i + 1)*p^(e - i), {i, 0, e}]; f[2, e_] := e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

PARI

a(n)={sumdiv(n, d, sigma(d>>valuation(d,2)))} \\ Andrew Howroyd, Jul 27 2018

Formula

L.g.f.: log(Product_{k>=1} (1 + x^k)^(sigma(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jun 19 2018

Multiplicative with a(2^e) = e+1 and a(p^e) = Sum_{i=0..e} (i+1)*p^(e-i) for e >= 0 and prime p > 2. - Werner Schulte, Jan 05 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/144 = 0.676452... . - Amiram Eldar, Oct 22 2022