A124316 a(n) = Sum_{d|n} sigma(gcd(d,n/d)), where sigma is the sum of divisors function, A000203.

1, 2, 2, 5, 2, 4, 2, 8, 6, 4, 2, 10, 2, 4, 4, 15, 2, 12, 2, 10, 4, 4, 2, 16, 8, 4, 10, 10, 2, 8, 2, 22, 4, 4, 4, 30, 2, 4, 4, 16, 2, 8, 2, 10, 12, 4, 2, 30, 10, 16, 4, 10, 2, 20, 4, 16, 4, 4, 2, 20, 2, 4, 12, 37, 4, 8, 2, 10, 4, 8, 2, 48, 2, 4, 16, 10, 4, 8, 2, 30, 23, 4, 2, 20, 4, 4, 4, 16, 2, 24

Offset: 1

Author

Robert G. Wilson v, Sep 30 2006

Keywords

• nonn
• easy
• mult

Comments

Apparently multiplicative and the inverse Mobius transform of A069290. - R. J. Mathar, Feb 07 2011

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Ekkehard Krätzel, Werner Georg Nowak, and László Tóth, On certain arithmetic functions involving the greatest common divisor, Cent. Eur. J. Math., 10 (2012), 761-774.
• Manfred Kühleitner and Werner Georg Nowak, On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions, Cent. Eur. J. Math., Vol. 11, No. 3 (2013), pp. 477-486; arXiv preprint, arXiv:1204.1146 [math.NT], 2012.
• László Tóth, Multiplicative arithmetic functions of several variables: a survey, in: T. Rassias, P. Pardalos (eds.) Mathematics Without Boundaries, Springer, New York, NY, 2024; arXiv preprint, arXiv:1310.7053 [math.NT], 2013-2014.

Crossrefs

Cf. A000203, A001616, A001620, A069290, A061884, A082633, A124315.

Programs

Maple

A124316 := proc(n) local a,d;
  a := 0 ;
  for d in numtheory[divisors](n) do
     igcd(d,n/d) ;
     a := a+numtheory[sigma](%) ;
   end do:
   a;
end proc: # R. J. Mathar, Apr 14 2011

Mathematica

Table[Plus @@ Map[DivisorSigma[1, GCD[ #, n/# ]] &, Divisors@n], {n, 90}]
f[p_, e_] := (If[OddQ[e], 2*p^((e+3)/2), p^(e/2 + 1)*(p+1)] - (e+3)*p + e + 1)/(p-1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 28 2024 *)

PARI

a(n) = sumdiv(n, d, sigma(gcd(d, n/d))); \\ Michel Marcus, Feb 13 2016

Python

from sympy import divisors, divisor_sigma, gcd
def a(n): return sum([divisor_sigma(gcd(d, n/d)) for d in divisors(n)]) # Indranil Ghosh, May 25 2017

Formula

From Amiram Eldar, Mar 28 2024: (Start)

Multiplicative with a(p^e) = (p^(e/2 + 1)*(p+1) - (e+3)*p + e + 1)/(p-1)^2, if e is even, and (2*p^((e+3)/2) - (e+3)*p + e + 1)/(p-1)^2 if e is odd.

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

Sum_{k=1..n} a(k) = (log(n)^2/4 + (2*gamma - 1/2)*log(n) + 5*gamma^2/2 - 2*gamma - 3*gamma_1 + 1/2) * n + O(n^(2/3)*log(n)^(16/9)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633) (Krätzel et al., 2012). (End)