A375032 - OEIS

%I #12 Aug 17 2024 09:16:25

%S 0,1,1,0,1,1,1,3,0,1,1,1,1,1,1,0,1,1,1,1,1,1,1,3,0,1,3,1,1,1,1,5,1,1,

%T 1,0,1,1,1,3,1,1,1,1,1,1,1,1,0,1,1,1,1,3,1,3,1,1,1,1,1,1,1,0,1,1,1,1,

%U 1,1,1,3,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,3,1,1,1,1,1,1,1,5,1,1,1,0,1,1,1,3,1

%N The maximum odd exponent in the prime factorization of n, or 0 if no such exponent exists.

%C The asymptotic density of the occurrences of 0's is 0 (the asymptotic density of squares).

%C The asymptotic density of the occurrences of 1's is d(0) = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465, asymptotic density of A335275).

%C The asymptotic density of the occurrences of 2*k+1, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+2)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k)*(p+1))).

%H Amiram Eldar, <a href="/A375032/b375032.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 max(a(n), A375033(n)) = A051903(n).

%F a(n) = 0 if and only if n is a square (A000290).

%F a(n) = 1 if and only if n is in A335275 \ A000290.

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} (2*k+1) * d(k) = 1.30000522546018852138..., where d(k) is defined in the Comments section above.

%F a(n) = A051903(A350389(n)). - _Amiram Eldar_, Aug 17 2024

%t a[n_] := Max[0, Max[Select[FactorInteger[n][[;; , 2]], OddQ]]]; a[1] = 0; Array[a, 100]

%o (PARI) a(n) = {my(e = select(x -> (x % 2), factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

%Y Cf. A000290, A051903, A065465, A162642, A335275, A350389, A375033.

%K nonn,easy

%O 1,8

%A _Amiram Eldar_, Jul 28 2024