A295295 Sum of squarefree divisors of the powerful part of n: a(n) = A048250(A057521(n)).

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 8, 4, 18, 1, 1, 1, 3, 1

Offset: 1

Author

Antti Karttunen, Nov 25 2017

Keywords

• nonn
• easy
• mult

Comments

The sum of the squarefree divisors of n whose square divides n. - Amiram Eldar, Oct 13 2023

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A000203, A003557, A048250, A057521, A064549, A092261, A295294, A344137.

Cf. A001620, A013661, A073002.

Programs

Mathematica

Array[DivisorSum[#/Denominator[#/Apply[Times, FactorInteger[#][[All, 1]]]^2], # &, SquareFreeQ] &, 105] (* Michael De Vlieger, Nov 26 2017, after Jean-François Alcover at A057521 *)
f[p_, e_] := If[e == 1, 1, p+1] ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)

PARI

a(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,2]==1, f[i,1]=1)); sumdiv(factorback(f), d, d*issquarefree(d)); \\ Michel Marcus, Jan 29 2021

Formula

Multiplicative with a(p) = 1 and a(p^e) = (p+1) for e > 1.

a(n) = A048250(A057521(n)) = A048250(A064549(A003557(n))).

a(n) = A048250(n) / A092261(n).

a(n) = Sum_{d^2|n} d * mu(d)^2. - Wesley Ivan Hurt, Feb 13 2022

From Amiram Eldar, Sep 18 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-1) / zeta(4*s-2).

Sum_{k=1..n} a(k) ~ (3*n/Pi^2) * (log(n) + 3*gamma - 1 - 4*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). (End)

a(n) = A048250(n) - A344137(n). - Amiram Eldar, Oct 13 2023