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A250031 a(n) is the numerator of the density of natural numbers m such that gcd(m,floor(m/n))=1. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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).

lim(n->infinity)a(n)/A250033(n) = 1/zeta(2) = A059956.

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

Cf. A059956, A248499, A248501, A250032, A250033, A250034.

Sequence in context: A010216 A166522 A206610 * A066552 A206609 A281085

Adjacent sequences:  A250028 A250029 A250030 * A250032 A250033 A250034

KEYWORD

nonn,frac

AUTHOR

Stanislav Sykora, Nov 16 2014

STATUS

approved

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Last modified June 28 16:56 EDT 2017. Contains 288839 sequences.