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

%I A070999 #28 Jun 29 2022 02:57:30
%S A070999 6,15,18,21,30,33,35,42,44,45,48,51,54,60,66,69,70,78,84,87,90,99,102,
%T A070999 105,114,119,120,123,126,132,133,135,138,140,141,144,147,150,153,159,
%U A070999 162,165,168,174,177,180,186,195,198,204,207,210,213,217,220,221,222
%N A070999 Numbers n such that the denominator of Sum_{k=1..n} 1/gcd(n,k) is not equal to n.
%C A070999 Does lim_{n->infinity} a(n)/n = 3?
%C A070999 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
%C A070999 Numbers k such that gcd(k, A057660(k)) > 1. - _Amiram Eldar_, Jun 29 2022
%H A070999 Amiram Eldar, <a href="/A070999/b070999.txt">Table of n, a(n) for n = 1..10000</a>
%e A070999 Sum_{k=1..6} 1/gcd(6,k) = 7/2, hence 6 is in the sequence;
%e A070999 Sum_{k=1..12} 1/gcd(12,k) = 77/12 so 12 is not in the sequence.
%t A070999 Select[Range[300],Denominator[Sum[1/GCD[#,k],{k,#}]]!=#&] (* _Harvey P. Dale_, May 07 2022 *)
%t A070999 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 *)
%o A070999 (PARI) for(n=1,300,if(denominator(sum(i=1,n,1/gcd(n,i)))<n,print1(n,",")))
%Y A070999 Cf. A000010, A018804, A057660.
%K A070999 easy,nonn
%O A070999 1,1
%A A070999 _Benoit Cloitre_, May 18 2002