OFFSET
1,4
COMMENTS
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).
LINKS
Stanislav Sykora, Table of n, a(n) for n = 1..1000
S. Sykora, On some number densities related to coprimes, Stan's Library, Vol.V, Nov 2014, DOI: 10.3247/SL5Math14.005
FORMULA
EXAMPLE
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.
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.
When n=1, only m=1 qualifies, so that the density is 0, and a(1) = 0.
PROG
(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)};
s(n)=1+s_aux(n, 1, 1);
a=vector(1000, n, numerator(1-s(n-1)/n))
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Stanislav Sykora, Nov 16 2014
STATUS
approved