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

Author

Amiram Eldar, Jun 24 2025

Keywords

• nonn
• easy

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

Crossrefs

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

Programs

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]

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))));

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...,