A288418 a(n) = Sum_{d|n} d^2*A000593(n/d).

1, 5, 13, 21, 31, 65, 57, 85, 130, 155, 133, 273, 183, 285, 403, 341, 307, 650, 381, 651, 741, 665, 553, 1105, 806, 915, 1210, 1197, 871, 2015, 993, 1365, 1729, 1535, 1767, 2730, 1407, 1905, 2379, 2635, 1723, 3705, 1893, 2793, 4030, 2765, 2257, 4433, 2850, 4030

Offset: 1

Author

Seiichi Manyama, Jun 09 2017

Keywords

• nonn
• mult

Comments

Multiplicative because this sequence is the Dirichlet convolution of A000290 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. A000290, A001001, A192065, A183699.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), this sequence (k=2), A288419 (k=3), A288420 (k=4).

Programs

Mathematica

a[n_] := DivisorSum[n, Function[d, d^2*DivisorSum[n/d, If[OddQ[#], #, 0]&]] ];
Array[a, 50] (* Jean-François Alcover, Jul 03 2017 *)
f[p_, e_] := (p^(e + 1) - 1)*(p^(e + 2) - 1)/((p - 1)*(p^2 - 1)); f[2, e_] := (4^(e + 1) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

PARI

a(n) = sumdiv(n, d, d^2*sigma((n/d)>>valuation(n/d, 2))); \\ Michel Marcus, Jul 03 2017; corrected Jun 12 2022

Formula

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

From Amiram Eldar, Nov 13 2022: (Start)

a(n) = A001001(n) for odd n.

Multiplicative with a(2^e) = (4^(e+1)-1)/3 and a(p^e) = (p^(e+1)-1)*(p^(e+2)-1)/((p-1)*(p^2-1)) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)*zeta(3)/4 = A183699 / 4 = 0.494326... . (End)