A057670 a(n) = Sum_{k|n} lcm(k, n/k).

1, 4, 6, 10, 10, 24, 14, 24, 21, 40, 22, 60, 26, 56, 60, 52, 34, 84, 38, 100, 84, 88, 46, 144, 55, 104, 72, 140, 58, 240, 62, 112, 132, 136, 140, 210, 74, 152, 156, 240, 82, 336, 86, 220, 210, 184, 94, 312, 105, 220, 204, 260, 106, 288, 220, 336, 228, 232, 118, 600

Offset: 1

Author

Leroy Quet, Oct 18 2000

Keywords

• nonn
• mult
• easy

Examples

			a(8) = lcm(1,8) + lcm(2,4) + lcm(4,2) + lcm(8,1) = 8 + 4 + 4 + 8 = 24.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv:1310.7053 [math.NT], 2013-2014.

Crossrefs

Cf. A001620, A055155, A107661.

Programs

Mathematica

Table[DivisorSum[n, LCM[#, n/#] &], {n, 59}] (* Michael De Vlieger, Dec 11 2017 *)
f[p_, e_] := (2*p^(e + 1) - p^Ceiling[(e + 1)/2] - p^Floor[(e + 1)/2])/(p - 1); f[p_, 1] := 2*p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Aug 27 2023 *)

PARI

a(n) = sumdiv(n, d, lcm(d, n/d)); \\ Michel Marcus, May 19 2014

Formula

Multiplicative with a(p) = 2*p, a(p^k) = (2*p^(k+1) - p^ceiling((k+1)/2) - p^floor((k+1)/2)) / (p-1). a(n) is odd iff n is an odd square. - Henry Bottomley, May 16 2005

Multiplicative with a(p^e) = Sum_{k=0..e} p^max(k, e-k), (cf. A107661). - Mitch Harris, May 18 2005

Dirichlet g.f.: (zeta(s-1))^2*zeta(2s-1)/zeta(2s-2). - R. J. Mathar, Feb 11 2011

Sum_{k=1..n} a(k) ~ 3*zeta(3)*n^2 / (2*Pi^2) * (2*log(n) - 24*zeta'(2)/Pi^2 - 1 + 4*gamma + 4*zeta'(3)/zeta(3)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 01 2019