A375032 - OEIS

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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

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OFFSET

1,8

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.

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

Sequence in context: A343858 A065714 A110700 * A338939 A292150 A181875

Adjacent sequences: A375029 A375030 A375031 * A375033 A375034 A375035

KEYWORD

nonn,easy

AUTHOR

Amiram Eldar, Jul 28 2024

STATUS

approved