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.

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

Author

Amiram Eldar, Jul 28 2024

Keywords

• sign
• easy

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).

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. A000302, A081294, A368714, A375032, A375033.

Programs

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]

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));}

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))).