A124315 a(n) = Sum_{ d divides n } tau(gcd(d,n/d)), where tau = sigma_0 = A000005.

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 8, 2, 4, 4, 9, 2, 8, 2, 8, 4, 4, 2, 12, 4, 4, 6, 8, 2, 8, 2, 12, 4, 4, 4, 16, 2, 4, 4, 12, 2, 8, 2, 8, 8, 4, 2, 18, 4, 8, 4, 8, 2, 12, 4, 12, 4, 4, 2, 16, 2, 4, 8, 16, 4, 8, 2, 8, 4, 8, 2, 24, 2, 4, 8, 8, 4, 8, 2, 18, 9, 4, 2, 16, 4, 4, 4, 12, 2, 16, 4, 8, 4, 4, 4, 24, 2, 8

Offset: 1

Author

Robert G. Wilson v, Sep 30 2006

Keywords

• mult
• nonn

Comments

Apparently the Mobius transform of A046951. - R. J. Mathar, Feb 07 2011

Number of ordered pairs of divisors of n, (d1,d2), with d1<=d2, such that d1|d2 and n|(d1*d2). - Wesley Ivan Hurt, Mar 22 2022

Links

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

Crossrefs

Cf. A000005, A001616, A001620, A046951, A061884, A124316, A306016.

Programs

Maple

A124315 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do igcd(d,n/d) ; a := a+numtheory[tau](%) ; end do: a; end proc: # R. J. Mathar, Apr 14 2011

Mathematica

Table[Plus @@ Map[DivisorSigma[0, GCD[ #, n/# ]] &, Divisors@n], {n, 98}]
f[p_, e_] := e + 1 + Floor[e^2/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 10 2022 *)

PARI

a(n) = sumdiv(n, d, numdiv(gcd(d, n/d))); \\ Michel Marcus, Feb 12 2016

Python

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

Formula

a(p) = 2 iff p is a prime.

Multiplicative with a(p^e) = e+1+floor(e^2/4). - R. J. Mathar, Apr 14 2011

Dirichlet g.f.: zeta(s)^2 * zeta(2*s). - Vaclav Kotesovec, Jan 11 2019

Sum_{k=1..n} a(k) ~ (Pi^2/6) * (n*log(n) + (2*gamma - 1 + 2*zeta'(2)/zeta(2))*n), where gamma is Euler's constant (A001620). - Amiram Eldar, Oct 22 2022