A071000 Numbers m such that the denominator of Sum_{k=1..n} 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

1,2

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

Links

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000010, A018804.

Sequence in context: A095410 A022293 A183218 * A088451 A047595 A079298

Adjacent sequences: A070997 A070998 A070999 * A071001 A071002 A071003

Keyword

easy,nonn

Author

Benoit Cloitre, May 18 2002

Status

approved