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

1,2

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.

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A257090 A360910 A339664 * A328705 A351346 A061034

Adjacent sequences: A124313 A124314 A124315 * A124317 A124318 A124319

Keyword

nonn,easy,mult

Author

Robert G. Wilson v, Sep 30 2006

Status

approved