login
A375032
The maximum odd exponent in the prime factorization of n, or 0 if no such exponent exists.
4
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
1,8
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))).
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)); }
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Jul 28 2024
STATUS
approved