A385130 The sum of divisors of n whose maximum exponent in their prime factorization is even.

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 17, 1, 1, 1, 21, 1, 28, 1, 25, 1, 1, 1, 17, 26, 1, 10, 33, 1, 1, 1, 21, 1, 1, 1, 80, 1, 1, 1, 25, 1, 1, 1, 49, 55, 1, 1, 81, 50, 76, 1, 57, 1, 28, 1, 33, 1, 1, 1, 97, 1, 1, 73, 85, 1, 1, 1, 73, 1, 1, 1, 80, 1, 1, 101, 81, 1, 1

Offset

1,4

Comments

The sum of terms in A368714 that divide n.

The number of these divisors is A385128(n).

Links

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

Formula

a(n) = Sum_{d|n} (d * (1 - A051903(d) mod 2)).

a(n) = A000203(n) - A385131(n).

a(n) = Sum_{k=1..kmax(n)} (-1)^(k+1) * Product_{i=1..r} (p_i^(min(e_i, k-1) + 1)-1)/(p_i-1), for n >= 2; if n = Product_{i=1..r} p_i^e_i, r = omega(n) = A001221(n), then emax(n) = max(e_i) = A051903(n), and kmax(n) = emax(n) if emax(n) is odd, and emax(n)+1 otherwise.

Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Sum_{k>=2} (-1)^k * (1-1/zeta(k)) = 0.27591672059822700769...,

Mathematica

q[n_] := EvenQ[Max[FactorInteger[n][[;; , 2]]]]; q[1] = True; a[n_] := DivisorSum[n, # &, q[#] &]; Array[a, 100]
(* second program: *)
a[n_] := Module[{f = FactorInteger[n], p, e, emax, kmax}, p = f[[;; , 1]]; e = f[[;; , 2]]; emax = Max[e]; kmax = emax + 1 - Mod[emax, 2]; Sum[(-1)^(k+1) * Product[(p[[i]]^(Min[e[[i]], k-1]+1)-1)/(p[[i]]-1), {i, 1, Length[e]}], {k, 1, kmax}]]; a[1] = 1; Array[a, 100]

Prog

(PARI) q(n) = if(n == 1, 1, !(vecmax(factor(n)[, 2]) % 2));
a(n) = sumdiv(n, d, d*q(d));
(PARI) a(n) = if(n == 1, 1, my(f = factor(n), p = f[, 1], e = f[, 2], emax = vecmax(e), kmax = emax + 1 - emax % 2); sum(k = 1, kmax, (-1)^(k+1) * prod(i = 1, #e, (p[i]^(min(e[i], k-1)+1)-1)/(p[i]-1))));

Crossrefs

Cf. A000203, A001221, A013661, A051903, A368714, A383156, A385128, A385131.

Sequence in context: A365403 A373439 A035316 * A293718 A068316 A388908

Adjacent sequences: A385127 A385128 A385129 * A385131 A385132 A385133

Keyword

nonn,easy

Author

Amiram Eldar, Jun 24 2025

Status

approved