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%I #18 Dec 02 2014 16:32:24
%S 0,1,1,13,8,26,19,163,361,1223,1307,16477,5749,83977,88267,280817,
%T 147916,1377406,2839897,58552633,60492571,63263911,65468386,403117367,
%U 549883871,579629587,596790577,1864736021,1912541636,29293503812,59449633388,969992016739
%N a(n) is the numerator of the density of natural numbers m such that gcd(m,floor(m/n))=1.
%C For introduction, see the comments in A250032. The present sequence is obtained when the condition P(m) is identified, for each chosen n>0, with the equality gcd(m,floor(m/n))=1, i.e., P(m)=1 when the equality holds, while P(m)=0 when it does not. Again, the densities d(n) exist and are rational numbers. The value of a(n) is the numerator of d(n), while A250033(n) is the denominator of d(n).
%H Stanislav Sykora, <a href="/A250031/b250031.txt">Table of n, a(n) for n = 1..1000</a>
%H S. Sykora, On some number densities related to coprimes, Stan's Library, Vol.V, Nov 2014, DOI: <a href="http://dx.doi.org/10.3247/SL5Math14.005">10.3247/SL5Math14.005</a>
%F For n>1, a(n)=A250033(n)-A250032(n), and a(n)/A250033(n)=1-s(n-1)/n, where s(n) A250034(n)/A072155(n).
%F lim(n->infinity)a(n)/A250033(n) = 1/zeta(2) = A059956.
%e When n=10, the density of numbers m that are coprime to floor(m/10) turns out to be 1223/2100. Hence a(10) = 1223/2100.
%e When n=2, all odd numbers qualify, but only the m=2 among even numbers does; hence the density is 1/2 and therefore a(2)=1.
%e When n=1, only m=1 qualifies, so that the density is 0, and a(1) = 0.
%o (PARI) s_aux(n,p0,inp)={my(t=0/1,tt=0/1,in=inp,pp);while(1,pp=p0*prime(in);tt=n\pp;if(tt==0,break,t+=tt/pp-s_aux(n,pp,in++)));return(t)};
%o s(n)=1+s_aux(n,1,1);
%o a=vector(1000,n,numerator(1-s(n-1)/n))
%Y Cf. A059956, A248499, A248501, A250032, A250033, A250034.
%K nonn,frac
%O 1,4
%A _Stanislav Sykora_, Nov 16 2014
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