OFFSET
1,2
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
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved