A385131 The sum of divisors of n whose maximum exponent in their prime factorization is odd.

0, 2, 3, 2, 5, 11, 7, 10, 3, 17, 11, 11, 13, 23, 23, 10, 17, 11, 19, 17, 31, 35, 23, 43, 5, 41, 30, 23, 29, 71, 31, 42, 47, 53, 47, 11, 37, 59, 55, 65, 41, 95, 43, 35, 23, 71, 47, 43, 7, 17, 71, 41, 53, 92, 71, 87, 79, 89, 59, 71, 61, 95, 31, 42, 83, 143, 67, 53

Offset: 1

Author

Amiram Eldar, Jun 24 2025

Keywords

• nonn
• easy

Comments

The sum of divisors of n that are not terms in A368714.

The number of these divisors is A385129(n).

Links

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

Crossrefs

Cf. A000203, A001221, A013661, A051903, A368714, A383156, A385129, A385130.

Programs

Mathematica

q[n_] := OddQ[Max[FactorInteger[n][[;; , 2]]]]; q[1] = False; 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 + Mod[emax, 2]; Sum[(-1)^k * Product[(p[[i]]^(Min[e[[i]], k-1]+1)-1)/(p[[i]]-1), {i, 1, Length[e]}], {k, 1, kmax}]]; a[1] = 0; Array[a, 100]

PARI

q(n) = if(n == 1, 0, vecmax(factor(n)[,2]) % 2);
a(n) = sumdiv(n, d, d*q(d));

PARI

a(n) = if(n == 1, 0, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), kmax = emax + emax % 2); sum(k = 1, kmax, (-1)^k * prod(i = 1, #e, (p[i]^(min(e[i], k-1)+1)-1)/(p[i]-1))));

Formula

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

a(n) = A000203(n) - A385130(n).

a(n) = Sum_{k=1..kmax(n)} (-1)^k * 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)+1 if emax(n) is odd, and emax(n) otherwise.

Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = 1 - Sum_{k>=2} (-1)^k * (1-1/zeta(k)) = 0.7240832794017729923099...,