OFFSET
1,2
COMMENTS
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
KEYWORD
nonn,mult
AUTHOR
Paul D. Hanna, Jul 03 2009
STATUS
approved