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A250031
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a(n) is the numerator of the density of natural numbers m such that gcd(m,floor(m/n))=1.
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6
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0, 1, 1, 13, 8, 26, 19, 163, 361, 1223, 1307, 16477, 5749, 83977, 88267, 280817, 147916, 1377406, 2839897, 58552633, 60492571, 63263911, 65468386, 403117367, 549883871, 579629587, 596790577, 1864736021, 1912541636, 29293503812, 59449633388, 969992016739
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OFFSET
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1,4
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COMMENTS
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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).
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LINKS
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S. Sykora, On some number densities related to coprimes, Stan's Library, Vol.V, Nov 2014, DOI: 10.3247/SL5Math14.005
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FORMULA
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EXAMPLE
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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.
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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