A309153 a(n) = A000203(n)*A001227(n).

1, 3, 8, 7, 12, 24, 16, 15, 39, 36, 24, 56, 28, 48, 96, 31, 36, 117, 40, 84, 128, 72, 48, 120, 93, 84, 160, 112, 60, 288, 64, 63, 192, 108, 192, 273, 76, 120, 224, 180, 84, 384, 88, 168, 468, 144, 96, 248, 171, 279, 288, 196, 108, 480, 288, 240, 320, 180, 120, 672, 124, 192, 624, 127, 336, 576, 136, 252, 384

Offset: 1

Author

Omar E. Pol, Jul 14 2019

Keywords

• nonn
• mult
• easy

Comments

A001227(n) is denoted by Delta_0(n) in Glaisher 1907.

a(n) = A000203(n) iff n is a power of 2.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
• Index entries for sequences mentioned by Glaisher.

Crossrefs

Cf. A000079, A000203, A001227, A062354, A062355, A064840.

Cf. A001620, A002117, A244115, A306016

Programs

Mathematica

Array[DivisorSum[#, 1 &, OddQ] DivisorSigma[1, #] &, 69] (* Michael De Vlieger, Nov 22 2019 *)
f[p_, e_] := (e+1)*(p^(e+1)-1)/(p-1); f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 01 2022 *)

Formula

a(n) = sigma(n)*delta(n).

Multiplicative with a(2^e) = 2^(e+1) - 1 and a(p^e) = (e+1)*(p^(e+1)-1)/(p-1) for p > 2. - Amiram Eldar, Nov 01 2022

From Amiram Eldar, Dec 04 2023: (Start)

Dirichlet g.f.: (4^s - 3*2^s + 2)/(4^s - 2) * (zeta(s)*zeta(s-1))^2/zeta(2*s-1).

Sum_{k=1..n} a(k) ~ (Pi^4/(168*zeta(3))) * n^2 * (log(n) + 2*gamma - 1/2 + 22*log(2)/21 + 2*zeta'(2)/zeta(2) - 2*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)