A385129 The number of divisors of n whose maximum exponent in their prime factorization is odd.

0, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 3, 1, 3, 3, 2, 1, 3, 1, 3, 3, 3, 1, 5, 1, 3, 2, 3, 1, 7, 1, 3, 3, 3, 3, 3, 1, 3, 3, 5, 1, 7, 1, 3, 3, 3, 1, 5, 1, 3, 3, 3, 1, 5, 3, 5, 3, 3, 1, 7, 1, 3, 3, 3, 3, 7, 1, 3, 3, 7, 1, 6, 1, 3, 3, 3, 3, 7, 1, 5, 2, 3, 1, 7, 3, 3, 3

Offset

1,6

Comments

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

The sum of these divisors is A385131(n).

Links

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

Formula

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

a(n) = A000005(n) - A385128(n).

a(n) = Sum_{k=1..kmax(n)} (-1)^k * Product_{i=1..r} (min(e_i, k-1) + 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 even, and emax(n)+1 otherwise.

Sum_{k=1..n} a(k) ~ c1 * n * (log(n) + 2*gamma - 1) - c2 * n, where gamma is Euler's constant (A001620), c1 = 1 - Sum_{k>=2} (-1)^k * (1-1/zeta(k)) = 0.7240832794017729923099..., and c2 = 1 + Sum_{k>=2} (-1)^k * k * zeta'(k)/zeta(k)^2 = 0.56812633046434345687... .

Mathematica

q[n_] := OddQ[Max[FactorInteger[n][[;; , 2]]]]; q[1] = False; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100]
(* second program: *)
a[n_] := Module[{e = FactorInteger[n][[;; , 2]], emax, kmax}, emax = Max[e]; kmax = emax + Mod[emax, 2]; Sum[(-1)^k * Product[Min[e[[i]], k-1] + 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, q(d));
(PARI) a(n) = if(n == 1, 0, my(e = factor(n)[, 2], emax = vecmax(e), kmax = emax + emax % 2); sum(k = 1, kmax, (-1)^k * prod(i = 1, #e, min(e[i], k-1)+1)));

Crossrefs

Cf. A000005, A001221, A001620, A051903, A368714, A383156, A385128, A385131.

Sequence in context: A157520 A325116 A227339 * A030777 A353375 A056595

Adjacent sequences: A385126 A385127 A385128 * A385130 A385131 A385132

Keyword

nonn,easy

Author

Amiram Eldar, Jun 24 2025

Status

approved