A375032 The maximum odd exponent in the prime factorization of n, or 0 if no such exponent exists.

0, 1, 1, 0, 1, 1, 1, 3, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 0, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 0, 1, 1, 1, 3, 1

Offset: 1

Author

Amiram Eldar, Jul 28 2024

Keywords

• nonn
• easy

Comments

The asymptotic density of the occurrences of 0's is 0 (the asymptotic density of squares).

The asymptotic density of the occurrences of 1's is d(0) = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465, asymptotic density of A335275).

The asymptotic density of the occurrences of 2*k+1, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+2)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k)*(p+1))).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000290, A051903, A065465, A162642, A335275, A350389, A375033.

Programs

Mathematica

a[n_] := Max[0, Max[Select[FactorInteger[n][[;; , 2]], OddQ]]]; a[1] = 0; Array[a, 100]

PARI

a(n) = {my(e = select(x -> (x % 2), factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

Formula

max(a(n), A375033(n)) = A051903(n).

a(n) = 0 if and only if n is a square (A000290).

a(n) = 1 if and only if n is in A335275 \ A000290.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} (2*k+1) * d(k) = 1.30000522546018852138..., where d(k) is defined in the Comments section above.

a(n) = A051903(A350389(n)). - Amiram Eldar, Aug 17 2024