A380325 The sum of the square roots of the squares that divide the n-th exponentially odd number.

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Author

Amiram Eldar, Jan 20 2025

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A013661, A065463, A069290, A268335, A330596, A380323, A380324.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], (p^((e+1)/2) - 1)/(p - 1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 0 &]

PARI

s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]^((f[i, 2]+1)/2) - 1)/(f[i, 1] - 1), 0));}
list(lim) = select(x -> x > 0, vector(lim, i, s(i)));

Formula

a(n) = A069290(A268335(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * A330596 / d = 1.74789521005721521109..., where d = A065463 is the asymptotic density of the exponentially odd numbers.