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

1,1

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

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A000010, A018804, A057660.

Sequence in context: A099535 A370920 A302296 * A128693 A105285 A138922

Adjacent sequences: A070996 A070997 A070998 * A071000 A071001 A071002

Keyword

easy,nonn

Author

Benoit Cloitre, May 18 2002

Status

approved