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A071000
Numbers m such that the denominator of Sum_{k=1..m} 1/gcd(m,k) equals m.
2
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
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
Sequence in context: A095410 A022293 A183218 * A088451 A047595 A079298
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, May 18 2002
STATUS
approved