A335852 Product of the exponents in the prime factorization of n in the ring of Gaussian integers.

1, 2, 1, 4, 1, 2, 1, 6, 2, 2, 1, 4, 1, 2, 1, 8, 1, 4, 1, 4, 1, 2, 1, 6, 4, 2, 3, 4, 1, 2, 1, 10, 1, 2, 1, 8, 1, 2, 1, 6, 1, 2, 1, 4, 2, 2, 1, 8, 2, 8, 1, 4, 1, 6, 1, 6, 1, 2, 1, 4, 1, 2, 2, 12, 1, 2, 1, 4, 1, 2, 1, 12, 1, 2, 4, 4, 1, 2, 1, 8, 4, 2, 1, 4, 1, 2

Offset: 1

Author

Amiram Eldar, Jun 26 2020

Keywords

• nonn
• mult

Comments

a(n) is also the number of divisors of n in Gaussian integers that are powerful (A335851).

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Gaussian Integer.
• Wikipedia, Gaussian integer.

Crossrefs

Cf. A005361, A062327, A335851.

Programs

Mathematica

a[n_] := Times @@ FactorInteger[n, GaussianIntegers -> True][[All, 2]]; Array[a, 100]

PARI

a(n) = my (f = factor(n*I)); f[1,1] /= I; prod(k=1, #f~, f[k,2]); \\ Michel Marcus, Jun 28 2020

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