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A375033
The maximum even exponent in the prime factorization of n, or 0 if no such exponent exists.
5
0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0
OFFSET
1,4
COMMENTS
First differs from A350386 at n = 36.
The asymptotic density of the occurrences of 0's is d(0) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463; the asymptotic density of the exponentially odd numbers, A268335).
The asymptotic density of the occurrences of 2*k, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+1)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k-1)*(p+1))).
For example, the asymptotic density of the occurrences of 2's is d(1) = Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = 0.243291... (the asymptotic density of A375031).
FORMULA
max(a(n), A375032(n)) = A051903(n).
a(n) = 0 if and only if n is an exponentially odd number (A268335).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (2*k) * d(k) = 0.72584606502990528747..., where d(k) is defined in the Comments section above.
a(n) = A051903(A350388(n)). - Amiram Eldar, Aug 17 2024
MATHEMATICA
a[n_] := Max[0, Max[Select[FactorInteger[n][[;; , 2]], EvenQ]]]; 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