cp's OEIS Frontend

A380323 The number of squares dividing the n-th exponentially odd number.

%I A380323 #8 Jan 21 2025 06:33:12
%S A380323 1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,3,1,1,1,1,1,1,2,1,1,1,
%T A380323 1,1,1,1,2,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,
%U A380323 3,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1
%N A380323 The number of squares dividing the n-th exponentially odd number.
%H A380323 Amiram Eldar, <a href="/A380323/b380323.txt">Table of n, a(n) for n = 1..10000</a>
%F A380323 a(n) = A046951(A268335(n)).
%F A380323 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)^2/A065463) * Product_{p prime} (1 - 3/p^2 + 2/p^3 + 1/p^4 - 1/p^5) = 1.18554061321703126489... .
%t A380323 f[p_, e_] := If[OddQ[e], (e + 1)/2, 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 0 &]
%o A380323 (PARI) s(n) = vecprod(apply(e -> if(e % 2, (e+1)/2, 0), factor(n)[, 2]));
%o A380323 list(lim) = select(x -> x > 0, vector(lim, i, s(i)));
%Y A380323 Cf. A013661, A046951, A065463, A268335, A380324, A380325.
%K A380323 nonn,easy
%O A380323 1,7
%A A380323 _Amiram Eldar_, Jan 20 2025