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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)



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).


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


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.


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.


(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))


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

Sequence in context: A303238 A166522 A206610 * A301825 A300377 A300679

Adjacent sequences:  A250028 A250029 A250030 * A250032 A250033 A250034




Stanislav Sykora, Nov 16 2014



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Last modified April 23 00:56 EDT 2021. Contains 343197 sequences. (Running on oeis4.)