A385129 -id:A385129 - cp's OEIS frontend

A385128 The number of divisors of n whose maximum exponent in their prime factorization is even.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 2, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 5, 2, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 3, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 3, 3, 1, 1, 1, 5, 3, 1, 1, 5, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 24 2025

The number of terms in A368714 that divide n.

The sum of these divisors is A385130(n).

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

Cf. A000005, A001221, A001620, A051903, A368714, A383156, A385129, A385130.

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

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

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

a(n) = Sum_{d|n} (1 - A051903(d) mod 2).
a(n) = A000005(n) - A385129(n).
a(n) = Sum_{k=1..kmax(n)} (-1)^(k+1) * 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 odd, 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 = Sum_{k>=2} (-1)^k * (1-1/zeta(k)) = 0.27591672059822700769..., and c2 = 1 + Sum_{k>=2} (-1)^k * k * zeta'(k)/zeta(k)^2 = 0.56812633046434345687... .

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

Original entry on oeis.org

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

Amiram Eldar, Jun 24 2025

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

The number of these divisors is A385129(n).

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

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

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]

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

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))));

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...,

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A385128 The number of divisors of n whose maximum exponent in their prime factorization is even.

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

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