A308084 a(n) = n*(n-1)*d(n)/4, where d(n)=A000005(n) is the number of divisors of n.

0, 1, 3, 9, 10, 30, 21, 56, 54, 90, 55, 198, 78, 182, 210, 300, 136, 459, 171, 570, 420, 462, 253, 1104, 450, 650, 702, 1134, 406, 1740, 465, 1488, 1056, 1122, 1190, 2835, 666, 1406, 1482, 3120, 820, 3444, 903, 2838, 2970, 2070, 1081, 5640, 1764, 3675, 2550

Offset

1,3

Comments

One eighth of row sums of A308805.

When n is not a square, d(n) is even; when n=k^2 for some k, n*(n-1)=(k-1)*k^2*(k+1) is a multiple of 4; in all cases a(n) is an integer.

Links

Luc Rousseau, Table of n, a(n) for n = 1..10000

Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Formula

a(n) = A000217(n-1)*A000005(n)/2.

From Peter Bala, Jan 21 2021: (Start)

G.f.: A(q) = (1/4)*Sum_{n >= 1} q^n*((n*2 + n)*q^n + n^2 - n)/(1 - q^n)^3.

Faster converging g.f.: A(q) = (1/4)*Sum_{n >= 1} q^(n^2)*( (n^2 + n)*q^n + n^2 - n)*( (n^2 - n)*q^(2*n) - 2*(n^2 - 1)*q^n + n^2 + n )/(1 - q^n)^3 - differentiate equation 5 in Arndt twice w.r.t. q and set x = 1. (End)

Mathematica

Table[n * (n - 1) * DivisorSigma[0, n] / 4, {n, 1, 50}] (* Amiram Eldar, Aug 14 2019 *)

Prog

(PARI)
for(n=1, 80, print1(n*(n-1)*numdiv(n)/4, ", "))

Crossrefs

Cf. A000005, A000217, A308805.

Sequence in context: A025616 A305091 A282635 * A136852 A031115 A004669

Adjacent sequences: A308081 A308082 A308083 * A308085 A308086 A308087

Keyword

nonn,easy

Author

Luc Rousseau, Aug 07 2019

Status

approved