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

1,2

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

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

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 *)

Prog

(Sage) A181797 = lambda n: n * sum(d for d in divisors(n) if is_squarefree(d)) # D. S. McNeil, Dec 05 2010
(PARI) a(n)=n*sumdiv(n, d, d*moebius(d)^2)

Crossrefs

Cf. A181798, A181799.

Sequence in context: A063722 A323058 A072294 * A185152 A028588 A327569

Adjacent sequences: A181794 A181795 A181796 * A181798 A181799 A181800

Keyword

nonn,easy,mult

Author

Matthew Vandermast, Dec 05 2010

Status

approved