A018806 Sum of gcd(x, y) for 1 <= x, y <= n.

1, 5, 12, 24, 37, 61, 80, 112, 145, 189, 220, 288, 325, 389, 464, 544, 593, 701, 756, 880, 989, 1093, 1160, 1336, 1441, 1565, 1700, 1880, 1965, 2205, 2296, 2488, 2665, 2829, 3028, 3328, 3437, 3621, 3832, 4152, 4273, 4621, 4748, 5040, 5373, 5597, 5736, 6168

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) is also the entrywise 1-norm of the n X n GCD matrix.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A018805, A064951, A344479.

Programs

Maple

N:= 1000 # to get a(1) to a(N)
g:= add(numtheory:-phi(k)*x^k*(1+x^k)/((1-x^k)^2*(1-x)),k=1..N):
S:= series(g, x, N+1):
seq(coeff(S,x,j), j=1..N); # Robert Israel, Jan 14 2015

Mathematica

Table[nn = n;Total[Level[Table[Table[GCD[i, j], {i, 1, nn}], {j, 1, nn}], {2}]], {n, 1, 48}] (* Geoffrey Critzer, Jan 14 2015 *)

PARI

a(n)=2*sum(i=1,n,sum(j=1,i-1,gcd(i,j)))+n*(n+1)/2 \\ Charles R Greathouse IV, Jun 21 2013

PARI

a(n)=sum(k=1,n,eulerphi(k)*(n\k)^2) \\ Charles R Greathouse IV, Jun 21 2013

Python

from sympy import totient
def A018806(n): return sum(totient(k)*(n//k)**2 for k in range(1,n+1)) # Chai Wah Wu, Aug 05 2024

Formula

Sum_{k=1..n} phi(k)*(floor(n/k))^2. - Vladeta Jovovic, Nov 10 2002

a(n) ~ kn^2 log n, with k = 6/Pi^2. - Charles R Greathouse IV, Jun 21 2013

G.f.: Sum_{k >= 1} phi(k)*x^k*(1+x^k)/((1-x^k)^2*(1-x)). - Robert Israel, Jan 14 2015