login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A250032 a(n) is the numerator of the density of natural numbers m such that gcd(m,floor(m/n))>1. 5
1, 1, 1, 11, 7, 19, 16, 117, 269, 877, 1003, 11243, 4261, 56163, 61883, 199663, 107339, 919889, 2009948, 38444267, 41354174, 43432679, 46078049, 266161243, 379669754, 387106183, 407127338, 1258564159, 1322304979, 19229195413, 40830611677, 634491904301, 2638247862269, 2717256540199, 2823435623209, 2886468920107, 1006725304509 (list; graph; refs; listen; history; text; internal format)
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
S. Sykora, On some number densities related to coprimes, Stan's Library, Vol. V, Nov 2014, DOI: 10.3247/SL5Math14.005
FORMULA
For n>1, a(n)/A250033(n) = s(n-1)/n, where s(n) = A250034(n)/A072155(n).
lim(n->infinity)a(n)/A250033(n) = 1-1/zeta(2) = A229099.
EXAMPLE
When n=1, S includes all natural numbers except 1, so d(1)=1. Hence a(1)=1 and A250033(1)=1.
When n=2, S includes all even numbers greater than 2, so d(2)=1/2. Hence a(2)=1 and A250033(2)=2.
When n=10, the subset S is A248500 and d(10)=877/2100. Hence a(10)=877 and A250033(10)=2100.
When n=16, S is A248502 and d(16)=199663/480480. Hence a(16)=199663 and A250033(16)=480480.
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
Sequence in context: A166521 A187866 A206419 * A305447 A060954 A350214
KEYWORD
nonn,frac
AUTHOR
Stanislav Sykora, Nov 16 2014
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 16 08:10 EDT 2024. Contains 374345 sequences. (Running on oeis4.)