A070999 Numbers n such that the denominator of Sum_{k=1..n} 1/gcd(n,k) is not equal to n.

6, 15, 18, 21, 30, 33, 35, 42, 44, 45, 48, 51, 54, 60, 66, 69, 70, 78, 84, 87, 90, 99, 102, 105, 114, 119, 120, 123, 126, 132, 133, 135, 138, 140, 141, 144, 147, 150, 153, 159, 162, 165, 168, 174, 177, 180, 186, 195, 198, 204, 207, 210, 213, 217, 220, 221, 222

Offset: 1

Author

Benoit Cloitre, May 18 2002

Keywords

• easy
• nonn

Comments

Does lim_{n->infinity} a(n)/n = 3?

Sum_{k=1..n} 1/gcd(n,k) = (1/n)*Sum_{d|n} phi(d)*d = (1/n)*Sum_{k=1..n} gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010. - Richard L. Ollerton, May 10 2021

Numbers k such that gcd(k, A057660(k)) > 1. - Amiram Eldar, Jun 29 2022

Examples

			Sum_{k=1..6} 1/gcd(6,k) = 7/2, hence 6 is in the sequence;
Sum_{k=1..12} 1/gcd(12,k) = 77/12 so 12 is not in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A018804, A057660.

Programs

Mathematica

Select[Range[300],Denominator[Sum[1/GCD[#,k],{k,#}]]!=#&] (* Harvey P. Dale, May 07 2022 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[250], !CoprimeQ[#, s[#]] &] (* Amiram Eldar, Jun 29 2022 *)

PARI

for(n=1,300,if(denominator(sum(i=1,n,1/gcd(n,i)))