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A375034
The difference between the maximum odd exponent and the maximum even exponent in the prime factorization of n, where 0 is assigned to each maximum exponent if no such exponent exists.
1
0, 1, 1, -2, 1, 1, 1, 3, -2, 1, 1, -1, 1, 1, 1, -4, 1, -1, 1, -1, 1, 1, 1, 3, -2, 1, 3, -1, 1, 1, 1, 5, 1, 1, 1, -2, 1, 1, 1, 3, 1, 1, 1, -1, -1, 1, 1, -3, -2, -1, 1, -1, 1, 3, 1, 3, 1, 1, 1, -1, 1, 1, -1, -6, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -3
OFFSET
1,4
COMMENTS
The indices of high value records are 1, 2, 8, 32, 128, 512, ... (A081294 with offset 1), and the indices of low value records are 1, 4, 16, 64, 256, 1024, ... (A000302 with offset 1).
FORMULA
a(n) = A375032(n) - A375033(n).
a(n) = 0 if and only if n = 1.
a(n) <= 0 if and only if n is in A368714.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (-1)^(k+1)*k*d(k) = 0.5741591604302832339078..., where d(k) = Product_{p prime} (1 - 1/(p^(k+1)*(p+1)) - Product_{p prime} (1 - 1/(p^(k-1)*(p+1)) for k >= 2, and d(1) = Product_{p prime} (1 - 1/(p^2*(p+1)).
MATHEMATICA
a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[0, Max[Select[e, OddQ]]] - Max[0, Max[Select[e, EvenQ]]]]; a[1] = 0; Array[a, 100]
PROG
(PARI) a(n) = {my(e = factor(n)[, 2], e1 = select(x -> (x % 2), e), e2 = select(x -> !(x % 2), e)); if(#e1 == 0, 0, vecmax(e1)) - if(#e2 == 0, 0, vecmax(e2)); }
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Amiram Eldar, Jul 28 2024
STATUS
approved