cp's OEIS Frontend

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.

%I A375034 #15 Mar 30 2025 04:33:34
%S A375034 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,
%T A375034 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,
%U A375034 1,1,-1,-6,1,1,1,-1,1,1,1,1,1,1,-1,-1,1,1,1,-3
%N 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.
%C A375034 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).
%H A375034 Amiram Eldar, <a href="/A375034/b375034.txt">Table of n, a(n) for n = 1..10000</a>
%H A375034 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A375034 a(n) = A375032(n) - A375033(n).
%F A375034 a(n) = 0 if and only if n = 1.
%F A375034 a(n) <= 0 if and only if n is in A368714.
%F A375034 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))).
%t A375034 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]
%o A375034 (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));}
%Y A375034 Cf. A000302, A081294, A368714, A375032, A375033.
%K A375034 sign,easy
%O A375034 1,4
%A A375034 _Amiram Eldar_, Jul 28 2024