A380325 -id:A380325 - cp's OEIS frontend

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

Amiram Eldar, Jan 20 2025

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

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

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

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)));

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... .

A380324 The sum of the squares dividing the n-th exponentially odd number.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 10, 1, 1, 1, 21, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 10, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5

Offset: 1

Amiram Eldar, Jan 20 2025

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

Cf. A035316, A268335, A380323, A380325.

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

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

a(n) = A035316(A268335(n)).

cp's OEIS Frontend

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

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