OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
V. C. Harris and M. V. Subbarao, On the divisor sum function, The Rocky Mountain Journal of Mathematics, Vol. 15, No. 2 (1985), pp. 399-412; alternative link.
FORMULA
Multiplicative with a(p^e) = A001249(e).
Dirichlet g.f.: (zeta(s)^8 / zeta(2*s)) * Product_{p prime} (1 + 8/p^s + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n * log(n)^15/15!, where c = (1/zeta(2)) * Product_{p prime} ((1 - 1/p)^8 * (1 + 8/p + 1/p^2)) = 0.00021468140977562218133... (Harris and Subbarao, 1985, Theorem 6.3, p. 406).
MATHEMATICA
f[p_, e_] := Binomial[e + 3, 3]^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecprod(apply(e -> binomial(e+3, 3)^2, factor(n)[, 2]));
(Python)
from math import prod, comb
from sympy import factorint
def A393214(n): return prod(comb(3+e, 3) for e in factorint(n).values())**2 # Chai Wah Wu, Feb 06 2026
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Amiram Eldar, Feb 06 2026
STATUS
approved
