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

1,2

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... .

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.

Formula

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

Example

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

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]

Prog

(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));

Crossrefs

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

Sequence in context: A133186 A084236 A068057 * A335852 A353751 A003484

Adjacent sequences: A376642 A376643 A376644 * A376646 A376647 A376648

Keyword

nonn,easy

Author

Amiram Eldar, Oct 01 2024

Status

approved