A376645 The maximum exponent in the factorization of n into powers of Gaussian primes.

0, 2, 1, 4, 1, 2, 1, 6, 2, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 6, 2, 2, 3, 4, 1, 2, 1, 10, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 4, 2, 2, 1, 8, 2, 2, 1, 4, 1, 3, 1, 6, 1, 2, 1, 4, 1, 2, 2, 12, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 2, 4, 1, 2, 1, 8, 4, 2, 1, 4, 1, 2, 1

Offset: 1

Author

Amiram Eldar, Oct 01 2024

Keywords

• nonn
• easy

Comments

a(n) = 0 only for n = 1. a(n) = k occurs infinitely many times for k >= 1. The numbers n = 2^e * m = 2^A007814(n) * A000265(n) for which a(n) = k and their asymptotic density are as follows:

1. k = 1: n is an odd squarefree number (A056911) and the density is d(1) = 2/(3*zeta(2)) = 0.405284... (A185199).

2. k >= 3 is odd: e < (k+1)/2 and m is a (k+1)-free number that is not a k-free number: d(k) = (1 - 1/2^((k+1)/2)) * (f(k+1)/zeta(k+1) - f(k)/zeta(k)), where f(k) = 1 - 1/2^k.

3. k >= 2 is even: e = k/2 and m is a (k+1)-free number, or e < k/2 and m is a (k+1)-free number that is not a k-free number: d(k) = (1/2^(k/2+1)) * f(k+1)/zeta(k+1) + (1-1/2^(k/2)) * (f(k+1)/zeta(k+1) - f(k)/zeta(k)), where f(k) is defined above.

The asymptotic mean of this sequence is Sum_{k>=1} k * d(k) = 2.64836785173193409440576... .

Examples

			a(2) = 2 because 2 = -i * (1+i)^2.
		

Links

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

Crossrefs

Cf. A000265, A051903, A007814, A056911, A078458, A086275, A185199, A335852, A375669.

Programs

Mathematica

a[n_] := Max[FactorInteger[n, GaussianIntegers -> True][[;; , 2]]]; a[1] = 0; Array[a, 100]
(* or *)
a[n_] := Module[{e = IntegerExponent[n, 2], od, em}, odd = n / 2^e; Max[2*e, If[odd == 1, 0, Max[FactorInteger[odd][[;;, 2]]]]]]; Array[a, 100]

PARI

a(n) = if(n == 1, 0, vecmax(factor(n*I)[, 2]));

PARI

a(n) = my(e = valuation(n, 2), es = factor(n >> e)[, 2]); max(2*e, if(#es, vecmax(es), 0));

Formula

a(n) = max(2*A007814(n), A051903(A000265(n))) = max(2*A007814(n), A375669(n)).