A375034 - OEIS

%I #12 Jul 28 2024 16:57:16

%S 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 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 1,1,-1,-6,1,1,1,-1,1,1,1,1,1,1,-1,-1,1,1,1,-3

%N 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 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 Amiram Eldar, <a href="/A375034/b375034.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%F a(n) = A375032(n) - A375033(n).

%F a(n) = 0 if and only if n = 1.

%F a(n) <= 0 if and only if n is in A368714.

%F 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 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 (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 Cf. A000302, A081294, A368714, A375032, A375033.

%K sign,easy

%O 1,4

%A _Amiram Eldar_, Jul 28 2024