A055155 a(n) = Sum_{d|n} gcd(d, n/d).

1, 2, 2, 4, 2, 4, 2, 6, 5, 4, 2, 8, 2, 4, 4, 10, 2, 10, 2, 8, 4, 4, 2, 12, 7, 4, 8, 8, 2, 8, 2, 14, 4, 4, 4, 20, 2, 4, 4, 12, 2, 8, 2, 8, 10, 4, 2, 20, 9, 14, 4, 8, 2, 16, 4, 12, 4, 4, 2, 16, 2, 4, 10, 22, 4, 8, 2, 8, 4, 8, 2, 30, 2, 4, 14, 8, 4, 8, 2, 20, 17, 4, 2, 16, 4, 4, 4, 12, 2, 20, 4, 8, 4, 4

Offset: 1

Author

Leroy Quet, Jul 02 2000

Keywords

• easy
• nonn
• mult

Comments

a(n) is odd iff n is odd square. - Vladeta Jovovic, Aug 27 2002

From Robert Israel, Dec 26 2015: (Start)

a(n) >= A000005(n), with equality iff n is squarefree (i.e., is in A005117).

a(n) = 2 iff n is prime. (End)

Examples

			a(9) = gcd(1,9) + gcd(3,3) + gcd(9,1) = 5, since 1, 3, 9 are the positive divisors of 9.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Ekkehard Krätzel, Werner Nowak and László Tóth, On certain arithmetic functions involving the greatest common divisor, Cent. Eur. J. Math., Vol. 10, No. 2 (2012), pp. 761-774.
• Manfred Kühleitner and Werner Georg Nowak, On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions, Central European Journal of Mathematics, Vol. 11, No. 3 (2013), pp. 477-486, preprint, arXiv: 1204.1146 [math.NT], 2012.
• Project Euler, Problem 530 - GCD of Divisors.
• László Tóth, Multiplicative arithmetic functions of several variables: a survey, in: T. Rassias and P. Pardalos (eds.), Mathematics Without Boundaries, Springer, New York, N.Y., 2014, pp. 483-514, arXiv preprint, arXiv:1310.7053 [math.NT], 2013-2014.

Crossrefs

Cf. A000005, A000188, A001620, A005117, A057670, A073002. A082633, A201994.

Cf. A000010.

Programs

Maple

N:= 1000: # to get a(1) to a(N)
V:= Vector(N):
for k from 1 to N do
   for j from 1 to floor(N/k) do
     V[k*j]:= V[k*j]+igcd(k,j)
   od
od:
convert(V,list); # Robert Israel, Dec 26 2015

Mathematica

Table[DivisorSum[n, GCD[#, n/#] &], {n, 94}] (* Michael De Vlieger, Sep 23 2017 *)
f[p_, e_] := If[EvenQ[e], (p^(e/2)*(p+1)-2)/(p-1), 2*(p^((e+1)/2)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 30 2020 *)

PARI

a(n) = sumdiv(n, d, gcd(d, n/d)); \\ Michel Marcus, Aug 03 2016

Python

from sympy import divisors, gcd
def A055155(n): return sum(gcd(d,n//d) for d in divisors(n,generator=True)) # Chai Wah Wu, Aug 19 2021

Formula

Multiplicative with a(p^e) = (p^(e/2)*(p+1)-2)/(p-1) for even e and a(p^e) = 2*(p^((e+1)/2)-1)/(p-1) for odd e. - Vladeta Jovovic, Nov 01 2001

Dirichlet g.f.: (zeta(s))^2*zeta(2s-1)/zeta(2s); inverse Mobius transform of A000188. - R. J. Mathar, Feb 16 2011

Dirichlet convolution of A069290 and A008966. - R. J. Mathar, Oct 31 2011

Sum_{k=1..n} a(k) ~ 3*n / (2*Pi^6) * (Pi^4 * log(n)^2 + ((8*g - 2)*Pi^4 - 24 * Pi^2 * z1) * log(n) + 2*Pi^4 * (1 - 4*g + 5*g^2 - 6*sg1) + 288 * z1^2 - 24 * Pi^2 * (-z1 + 4*g*z1 + z2)), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant A082633, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994. - Vaclav Kotesovec, Feb 01 2019

a(n) = (1/n)*Sum_{i=1..n} sigma(gcd(n,i^2)). - Ridouane Oudra, Dec 30 2020

a(n) = Sum_{k=1..n} gcd(gcd(n,k),n/gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010. - Richard L. Ollerton, May 09 2021