A071000 Numbers m such that the denominator of Sum_{k=1..m} 1/gcd(m,k) equals m.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 40, 41, 43, 46, 47, 49, 50, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 88, 89, 91, 92, 93

Offset: 1

Author

Benoit Cloitre, May 18 2002

Keywords

• easy
• nonn

Comments

Does lim_{n -> infinity} a(n)/n = 3/2?

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

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 9, 78, 709, 6713, 65135, 637603, 6275585, 61972835, 613362869, 6080312594, ... . Apparently, the asymptotic density of this sequence is 0 and the limit in the question above is infinite. - Amiram Eldar, Jun 28 2022

Examples

			Sum_{k=1..12} 1/gcd(12,k) = 77/12 hence 12 is in the sequence.
		

Links

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

Crossrefs

Cf. A000010, A018804.

Programs

Mathematica

Select[Range[100],Denominator[Sum[1/GCD[#,k],{k,#}]]==#&] (* Harvey P. Dale, Dec 13 2011 *)

PARI

for(n=1,300,if(denominator(sum(i=1,n,1/gcd(n,i))) == n,print1(n,",")))