OFFSET
1,2
COMMENTS
a(n) is also the number of divisors of n in Gaussian integers that are powerful (A335851).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gaussian Integer.
Wikipedia, Gaussian integer.
FORMULA
Multiplicative with a(p^e) = 2*e if p = 2, e if p == 3 (mod 4) and e^2 if p == 1 (mod 4).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (5/2) * Product_{p prime == 3 (mod 4)} (p^2 - p + 1)/(p*(p-1)) * Product_{p prime == 3 (mod 1)} (p^4 - 3*p^3 + 6*p^2 - 5*p + 1)/(p*(p-1)^3) = 3.73805905189... . - Amiram Eldar, Oct 15 2022
EXAMPLE
a(2) = 2 since 2 = -i * (1 + i)^2 and the Gaussian prime (1 + i) has an exponent 2.
a(100) = 16 since 100 = (1 + i)^4 * (1 + 2*i)^2 * (2 + i)^2 and 4*2*2 = 16.
MATHEMATICA
a[n_] := Times @@ FactorInteger[n, GaussianIntegers -> True][[All, 2]]; Array[a, 100]
PROG
(PARI) a(n) = my (f = factor(n*I)); f[1, 1] /= I; prod(k=1, #f~, f[k, 2]); \\ Michel Marcus, Jun 28 2020
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Amiram Eldar, Jun 26 2020
STATUS
approved