A181797 a(n) = n multiplied by the sum of its squarefree divisors (A048250(n)).

1, 6, 12, 12, 30, 72, 56, 24, 36, 180, 132, 144, 182, 336, 360, 48, 306, 216, 380, 360, 672, 792, 552, 288, 150, 1092, 108, 672, 870, 2160, 992, 96, 1584, 1836, 1680, 432, 1406, 2280, 2184, 720, 1722, 4032, 1892, 1584, 1080, 3312, 2256, 576, 392, 900

Offset: 1

Author

Matthew Vandermast, Dec 05 2010

Keywords

• nonn
• easy
• mult

Comments

Sum of reciprocals converges to Pi^2/6. The natural density of positive integers m such that A003557(m) = n equals 6/(a(n)*Pi^2).

If m is coprime to 6, a(3m) = a(4m).

Apparently the absolute values of the Dirichlet inverse of A000082. - R. J. Mathar, Mar 14 2011

Links

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

Crossrefs

Cf. A181798, A181799.

Programs

Maple

A181797 := proc(n) local f; f := ifactors(n)[2] ;  mul( op(1,d)^op(2,d)*( op(1,d)+1),d=f) ; end proc: # R. J. Mathar, Dec 05 2010

Mathematica

Table[n*Sum[d*MoebiusMu[d]^2, {d, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, Feb 02 2019 *)

PARI

a(n)=n*sumdiv(n,d,d*moebius(d)^2)

Sage

A181797 = lambda n: n * sum(d for d in divisors(n) if is_squarefree(d)) # D. S. McNeil, Dec 05 2010

Formula

a(n) = n*A048250(n). Multiplicative with a(p^e) = (p+1)*p^e.

Dirichlet g.f. zeta(s-1)*zeta(s-2)/zeta(2*s-4). - R. J. Mathar, Mar 14 2011

G.f.: x*f'(x), where f(x) = Sum_{k>=1} mu(k)^2*k*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 10 2017

Sum_{k=1..n} a(k) ~ n^3 / 3. - Vaclav Kotesovec, Feb 02 2019

Sum_{k>=1} 1/a(k) = Pi^2/6. - Vaclav Kotesovec, Sep 19 2020