A057661 a(n) = Sum_{k=1..n} lcm(n,k)/n.

1, 2, 4, 6, 11, 11, 22, 22, 31, 32, 56, 39, 79, 65, 74, 86, 137, 92, 172, 116, 151, 167, 254, 151, 261, 236, 274, 237, 407, 221, 466, 342, 389, 410, 452, 336, 667, 515, 550, 452, 821, 452, 904, 611, 641, 761, 1082, 599, 1051, 782, 956, 864, 1379, 821, 1166

Offset: 1

Author

Henry Gould, Oct 15 2000

Keywords

• easy
• nice
• nonn

Comments

Sum of numerators of n-th order Farey series (cf. A006842). - Benoit Cloitre, Oct 28 2002

Equals row sums of triangle A143613. - Gary W. Adamson, Aug 27 2008

Equals row sums of triangle A159936. - Gary W. Adamson, Apr 26 2009

Also row sums of triangle A164306. - Reinhard Zumkeller, Aug 12 2009

References

• H. W. Gould and Temba Shonhiwa, Functions of GCD's and LCM's, Indian J. Math. (Allahabad), 39 (1997), 11-35.
• H. W. Gould and Temba Shonhiwa, A generalization of Cesaro's function and other results, Indian J. Math. (Allahabad), 39 (1997), 183-194.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Zachary Franco, Problem 12114, The American Mathematical Monthly, Vol. 126, No. 5 (2019), p. 469; A Dirichlet Series with Reduced Numerators, Solution to Problem 12114 by Tamas Wiandt, ibid., Vol. 128, No. 1 (2021), pp. 91-92.
• Index entries for sequences related to lcm's.

Crossrefs

Cf. A000010, A000217, A006842, A018804, A023896, A051193, A057660, A143613, A159936, A164306.

See A341316 for another version.

Programs

Haskell

a057661 n = a051193 n `div` n  -- Reinhard Zumkeller, Jun 10 2015

Magma

[&+[&+[h: h in [1..d] | GCD(h,d) eq 1]: d in Divisors(n)]: n in [1..100]]; // Jaroslav Krizek, Dec 28 2016

Mathematica

Table[Total[Numerator[Range[n]/n]], {n, 55}] (* Alonso del Arte, Oct 07 2011 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; Array[a, 100] (* Amiram Eldar, Apr 26 2023 *)

PARI

a(n)=sum(k=1,n,lcm(n,k))/n \\ Charles R Greathouse IV, Feb 07 2017

Python

from math import lcm
def A057661(n): return sum(lcm(n,k)//n for k in range(1,n+1)) # Chai Wah Wu, Aug 24 2023

Python

from math import prod
from sympy import factorint
def A057661(n): return 1+prod((p**((e<<1)+1)+1)//(p+1) for p,e in factorint(n).items())>>1 # Chai Wah Wu, Aug 05 2024

Formula

a(n) = (1+A057660(n))/2.

a(n) = A051193(n)/n.

a(n) = Sum_{d|n} psi(d), where psi(m) = is the sum of totatives of m (A023896). - Jaroslav Krizek, Dec 28 2016

a(n) = Sum_{i=1..n} denominator(n/i). - Wesley Ivan Hurt, Feb 26 2017

G.f.: x/(2*(1 - x)) + (1/2)*Sum_{k>=1} k*phi(k)*x^k/(1 - x^k), where phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, Aug 31 2017

If p is prime, then a(p) = T(p-1) + 1 = p(p-1)/2 + 1, where T(n) = n(n+1)/2 is the n-th triangular number (A000217). - David Terr, Feb 10 2019

Sum_{k=1..n} a(k) ~ zeta(3) * n^3 / Pi^2. - Vaclav Kotesovec, May 29 2021

Dirichlet g.f.: zeta(s)*(1 + zeta(s-2)/zeta(s-1))/2 (Franco, 2019). - Amiram Eldar, Mar 26 2022

Extensions

More terms from James Sellers, Oct 16 2000