A061884 -id:A061884 - cp's OEIS frontend

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

Robert G. Wilson v, Sep 30 2006

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

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.

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

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

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 *)

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

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

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

A332713 a(n) = Sum_{d|n} phi(d/gcd(d, n/d)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 8, 10, 11, 12, 13, 14, 15, 13, 17, 16, 19, 20, 21, 22, 23, 21, 22, 26, 22, 28, 29, 30, 31, 24, 33, 34, 35, 32, 37, 38, 39, 35, 41, 42, 43, 44, 40, 46, 47, 39, 44, 44, 51, 52, 53, 44, 55, 49, 57, 58, 59, 60, 61, 62, 56, 46, 65, 66, 67, 68, 69, 70

Offset: 1

Ilya Gutkovskiy, Feb 20 2020

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A000010, A001616, A010052, A046790 (numbers n such that a(n) < n), A055653, A061884, A078779 (fixed points), A332619, A332686, A332712.

Table[Sum[EulerPhi[d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 70}]
A055653[n_] := Sum[Boole[GCD[d, n/d] == 1] EulerPhi[d], {d, Divisors[n]}]; a[n_] := Sum[Boole[IntegerQ[(n/d)^(1/2)]] A055653[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]

a(n) = sumdiv(n, d, eulerphi(d/gcd(d, n/d))); \\ Michel Marcus, Feb 20 2020

Dirichlet g.f.: zeta(s) * zeta(2*s) * zeta(s - 1) * Product_{p prime} (1 - p^(-s) + p^(-2*s) - p^(1 - 2*s)).
a(n) = Sum_{d|n} phi(lcm(d, n/d)/d).
a(n) = Sum_{d|n} A010052(n/d) * A055653(d).
Sum_{k=1..n} a(k) ~ c * Pi^6 * n^2 / 1080, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085... - Vaclav Kotesovec, Feb 22 2020
From Richard L. Ollerton, May 10 2021: (Start)
a(n) = Sum_{k=1..n} phi(gcd(n,k)/gcd(gcd(n,k),n/gcd(n,k)))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(n/gcd(n,k)/gcd(gcd(n,k),n/gcd(n,k)))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(lcm(gcd(n,k),n/gcd(n,k))/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(lcm(gcd(n,k),n/gcd(n,k))*gcd(n,k)/n)/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} A010052(gcd(n,k))*A055653(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} A010052(n/gcd(n,k))*A055653(gcd(n,k))/phi(n/gcd(n,k)). (End)

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

Original entry on oeis.org

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

Robert G. Wilson v, Sep 30 2006

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

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.

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

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

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 *)

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

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

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)

A374352 a(n) = [n>1] * a(n-1) + Sum_{d|n} phi(lcm(d,n/d)) where [] is an Iverson bracket.

Original entry on oeis.org

1, 3, 7, 12, 20, 28, 40, 52, 66, 82, 102, 122, 146, 170, 202, 228, 260, 288, 324, 364, 412, 452, 496, 544, 588, 636, 684, 744, 800, 864, 924, 980, 1060, 1124, 1220, 1290, 1362, 1434, 1530, 1626, 1706, 1802, 1886, 1986, 2098, 2186, 2278, 2382, 2472, 2560, 2688

Offset: 1

Alois P. Heinz, Jul 05 2024

nonn

Sum over all positive integers k, m with k*m <= n of phi(lcm(k,m)).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Iverson bracket

Partial sums of A061884.

Cf. A000010, A003990, A051173, A372866.

a:= proc(n) option remember; uses numtheory; `if`(n<1, 0,
      a(n-1)+add(phi(ilcm(d, n/d)), d=divisors(n)))
    end:
seq(a(n), n=1..66);

a(n) = Sum_{j=1..n} Sum_{d|j} phi(lcm(d,j/d)).

a(n) = Sum_{j=1..n} A061884(j).

cp's OEIS Frontend

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

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A332713 a(n) = Sum_{d|n} phi(d/gcd(d, n/d)).

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

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Maple

Mathematica

PARI

Python

Formula

A374352 a(n) = [n>1] * a(n-1) + Sum_{d|n} phi(lcm(d,n/d)) where [] is an Iverson bracket.

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