A162419 a(n) = sigma(n)*|A002129(n)| where sigma(n) = A000203(n).

1, 3, 16, 35, 36, 48, 64, 195, 169, 108, 144, 560, 196, 192, 576, 899, 324, 507, 400, 1260, 1024, 432, 576, 3120, 961, 588, 1600, 2240, 900, 1728, 1024, 3843, 2304, 972, 2304, 5915, 1444, 1200, 3136, 7020, 1764, 3072, 1936, 5040, 6084, 1728, 2304, 14384

Offset

1,2

Comments

A002129 forms the l.g.f. of log(Sum_{n>=0} x^(n(n+1)/2)), while A000203 forms the l.g.f. of log(1/eta(x)) where eta(x)^3 = Sum_{n>=0} (-1)^n*(2n+1)*x^(n*(n+1)/2).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(2n-1) = sigma(2n-1)^2.

L.g.f.: L(x) = log(G(x)) where G(x) is the g.f. of A162420.

From Amiram Eldar, Dec 01 2022: (Start)

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

Sum_{k=1..n} a(k) ~ c * n^3, where c = 29*zeta(3)/48 = 0.726242... . (End)

Dirichlet g.f.: (zeta(s)*zeta(s-1)^2*zeta(s-2)/zeta(2*s-2))*(7*2^(2-s)-4^(2-s)+2^s-4)/(2^s+2). - Amiram Eldar, Jan 06 2023

Example

L.g.f.: L(x) = x + 3*x^2/2 + 16*x^3/3 + 35*x^4/4 + 36*x^5/5 + 48*x^6/6 + ... where exp(L(x)) is the g.f. of A162420:
exp(L(x)) = 1 + x + 2*x^2 + 7*x^3 + 16*x^4 + 28*x^5 + 57*x^6 + ...
...
Equals the term-wise product of the (unsigned) sequences:
A000203:[1, 3,4, 7,6,12,8, 15,13,18,12, 28,14,24,24, 31,18,...];
A002129:[1,-1,4,-5,6,-4,8,-13,13,-6,12,-20,14,-8,24,-29,18,...].

Mathematica

f[p_, e_] := If[p == 2, (2^(e + 1) - 1) * (2^(e + 1) - 3), ((p^(e + 1) - 1)/(p - 1))^2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 48] (* Amiram Eldar, Jul 20 2019 *)

Prog

(PARI) a(n)=sigma(n)*sumdiv(n, d, (-1)^(n-d)*d)

Crossrefs

Cf. A162420, A000203, A002129, A002117.

Sequence in context: A196264 A031080 A013199 * A322191 A359559 A076153

Adjacent sequences: A162416 A162417 A162418 * A162420 A162421 A162422

Keyword

nonn,mult

Author

Paul D. Hanna, Jul 03 2009

Status

approved