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A366438 The number of divisors of the exponentially odd numbers (A268335).

%I A366438 #15 Oct 11 2023 18:23:42
%S A366438 1,2,2,2,4,2,4,4,2,2,4,4,2,2,4,4,2,8,4,4,2,8,2,6,4,4,4,2,4,4,8,2,8,2,
%T A366438 4,2,4,2,8,4,8,4,4,2,2,4,4,8,2,4,8,2,2,4,4,8,2,4,2,4,4,4,8,2,4,4,4,4,
%U A366438 12,2,2,8,2,8,8,4,2,2,8,4,2,8,4,4,4,16,4
%N A366438 The number of divisors of the exponentially odd numbers (A268335).
%C A366438 1 is the only odd term in this sequence.
%H A366438 Amiram Eldar, <a href="/A366438/b366438.txt">Table of n, a(n) for n = 1..10000</a>
%F A366438 a(n) = A000005(A268335(n)).
%t A366438 f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, OddQ], Times @@ (e + 1), Nothing]]; f[1] = 1; Array[f, 150]
%o A366438 (PARI) lista(max) = for(k = 1, max, my(e = factor(k)[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(vecprod(apply(x -> x+1, e)), ", ")));
%o A366438 (Python)
%o A366438 from math import prod
%o A366438 from itertools import count, islice
%o A366438 from sympy import factorint
%o A366438 def A366438_gen(): # generator of terms
%o A366438     for n in count(1):
%o A366438         f = factorint(n).values()
%o A366438         if all(e&1 for e in f):
%o A366438             yield prod(e+1 for e in f)
%o A366438 A366438_list = list(islice(A366438_gen(),30)) # _Chai Wah Wu_, Oct 10 2023
%Y A366438 Cf. A000005, A268335, A366439.
%Y A366438 Similar sequences: A048691, A072048, A076400, A358040, A363194, A363195.
%K A366438 nonn,easy
%O A366438 1,2
%A A366438 _Amiram Eldar_, Oct 10 2023