A102613 Numerator of the reduced fractions of the ratios of the number of primes less than n over the number of composites less than n.

0, 1, 2, 1, 3, 1, 4, 1, 4, 2, 5, 5, 6, 3, 2, 3, 7, 7, 8, 2, 8, 4, 9, 3, 9, 9, 1, 9, 10, 1, 11, 11, 1, 11, 11, 11, 12, 6, 4, 3, 13, 13, 14, 7, 14, 7, 15, 5, 15, 3, 5, 15, 16, 8, 16, 2, 16, 8, 17, 17, 18, 9, 2, 9, 18, 3, 19, 19, 19, 19, 20, 5, 21, 21, 7, 21, 3, 7, 22, 11, 22, 11, 23, 23, 23, 23

Offset: 1

Author

Cino Hilliard, Jan 30 2005

Keywords

• frac
• nonn
• easy

Comments

Conjecture: The ratio pi(x)/(n-pi(x)) tends to 0 as n tends to infinity. This is evident from the fact that Li(x)/(n-Li(x)) -> 0 as n -> infinity but unfortunately not proof.

Crossrefs

Cf. A000720, A062298.

Programs

PARI

pixovcmpx(n) = for(x=1,n,print1(numerator(pi(x)/(x-pi(x)))",")) pi(n) = \Number of primes less than or equal to n. { local(c,x); c=0;forprime(x=1,n,c++);return(c) }

PARI

a(n)=numerator(primepi(n)/(n-primepi(n))) \\ Jason Yuen, Aug 31 2024

Formula

a(n) = numerator(pi(n)/(n-pi(n))) = numerator(A000720(n)/A062298(n)). - Jason Yuen, Aug 31 2024