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

1,4

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).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

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

Cf. A000302, A081294, A368714, A375032, A375033.

Sequence in context: A327503 A051904 A070012 * A373365 A071178 A366895

Adjacent sequences: A375031 A375032 A375033 * A375035 A375036 A375037

Keyword

sign,easy

Author

Amiram Eldar, Jul 28 2024

Status

approved