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

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, 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, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset

1,7

Links

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

Formula

a(n) = A046951(A268335(n)).

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... .

Mathematica

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

Prog

(PARI) s(n) = vecprod(apply(e -> if(e % 2, (e+1)/2, 0), factor(n)[, 2]));
list(lim) = select(x -> x > 0, vector(lim, i, s(i)));

Crossrefs

Cf. A013661, A046951, A065463, A268335, A380324, A380325.

Sequence in context: A131789 A108465 A372467 * A367419 A069347 A161606

Adjacent sequences: A380320 A380321 A380322 * A380324 A380325 A380326

Keyword

nonn,easy,new

Author

Amiram Eldar, Jan 20 2025

Status

approved