OFFSET
1,4
COMMENTS
Let m be any natural number, and P(m) a relational expression on m (i.e., a property of m) evaluating to either 0 (false) or 1 (true). This defines a subset S of natural numbers N for which P(m)=1. When there exists a limit d=limit(M->infinity, Sum(m=1..M, P(m))/M), d is said to be the limit mean density (or just density) of the subset S in N. Now, choose an integer parameter n and set P(m)=gcd(m,floor(m/n))>1. This makes the property P, the corresponding subset S, and the density d all dependent upon n. The reference proves that for any n>0, the density d(n) exists and is a rational number. 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
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(s(n-1)/n))
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Stanislav Sykora, Nov 16 2014
STATUS
approved