A374459 - OEIS

%I #7 Jul 10 2024 09:37:50

%S 8,24,27,32,40,54,56,88,96,104,120,125,128,135,136,152,160,168,184,

%T 189,216,224,232,243,248,250,264,270,280,296,297,312,328,343,344,351,

%U 352,375,376,378,384,408,416,424,440,456,459,472,480,486,488,512,513,520,536

%N Nonsquarefree exponentially odd numbers.

%C First differs from A301517 at n = 1213. A301517(1213) = 12500 = 2^2 * 5^5 is not an exponentially odd number.

%C Numbers whose exponents in their prime factorization are all odd and at least one of them is larger than 1.

%C All the squarefree numbers (A005117) are exponentially odd. Therefore, the sequence of exponentially odd numbers (A268335) is a disjoint union of the squarefree numbers and this sequence.

%C The asymptotic density of this sequence is A065463 - A059956 = 0.096515099145... .

%H Amiram Eldar, <a href="/A374459/b374459.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A268335(A374460(n)).

%F Sum_{n>=1} 1/a(n)^s = zeta(2*s) * (Product_{p prime} (1 + 1/p^s - 1/p^(2*s))) - zeta(s)/zeta(2*s) for s > 1.

%e 8 = 2^3 is a term since 3 is odd and larger than 1.

%t q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, OddQ] && ! AllTrue[e, # == 1 &]]; Select[Range[1000], q]

%o (PARI) is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] %2), return(0))); for(i = 1, #e, if(e[i] >1, return(1))); 0;}

%Y Intersection of A013929 (or A046099) and A268335.

%Y Subsequence of A301517.

%Y Subsequences: A062838 \ {1}, A065036, A102838, A113850, A113852, A179671, A190011, A335988 \ {1}.

%Y Cf. A005117, A374460.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Jul 09 2024