A162419 - OEIS

%I #21 Jan 06 2023 20:46:01

%S 1,3,16,35,36,48,64,195,169,108,144,560,196,192,576,899,324,507,400,

%T 1260,1024,432,576,3120,961,588,1600,2240,900,1728,1024,3843,2304,972,

%U 2304,5915,1444,1200,3136,7020,1764,3072,1936,5040,6084,1728,2304,14384

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

%C 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).

%H Amiram Eldar, <a href="/A162419/b162419.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F From _Amiram Eldar_, Dec 01 2022: (Start)

%F 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.

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

%F 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

%e 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:

%e exp(L(x)) = 1 + x + 2*x^2 + 7*x^3 + 16*x^4 + 28*x^5 + 57*x^6 + ...

%e ...

%e Equals the term-wise product of the (unsigned) sequences:

%e A000203:[1, 3,4, 7,6,12,8, 15,13,18,12, 28,14,24,24, 31,18,...];

%e A002129:[1,-1,4,-5,6,-4,8,-13,13,-6,12,-20,14,-8,24,-29,18,...].

%t 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 *)

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

%Y Cf. A162420, A000203, A002129, A002117.

%K nonn,mult

%O 1,2

%A _Paul D. Hanna_, Jul 03 2009