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

1,2

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

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...,

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]

Prog

(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))));

Crossrefs

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

Sequence in context: A183098 A183101 A343300 * A285309 A250096 A345302

Adjacent sequences: A385128 A385129 A385130 * A385132 A385133 A385134

Keyword

nonn,easy

Author

Amiram Eldar, Jun 24 2025

Status

approved