A111114 - OEIS

%I #27 Dec 09 2024 10:11:05

%S 3,2,3,3,4,4,4,5,7,6,7,6,7,7,8,8,8,8,8,9,9,9,9,10,11,11,11,10,11,11,

%T 11,12,12,13,13,13,13,13,14,13,13,13,13,14,14,14,14,15,15,15,15,15,15,

%U 16,16,16,16,16,16,15,16,17,17,17,17,17,17,18,18,17,17,17,17,18,18,18,18

%N Integer part of prime(n)/pi(n).

%C Conjecture: As n -> infinity, there are infinitely many n's such that a(n) is greater than a(n+1).

%C a(n) > a(n + 1) only if n + 1 is prime. - _David A. Corneth_, Aug 22 2020

%H Harry J. Smith, <a href="/A111114/b111114.txt">Table of n, a(n) for n = 2..10000</a>

%e a(2) = 3 as prime(2) = 3, pi(2) = 1 so a(2) = floor(3/1) = 3.

%e a(3) = 2 as prime(3) = 5, pi(3) = 2 so a(3) = floor(5/2) = 2.

%t Table[IntegerPart[Prime[n]/PrimePi[n]],{n,2,80}] (* _Harvey P. Dale_, Nov 19 2014 *)

%o (PARI) a(n) = prime(n)\primepi(n)

%o (PARI) first(n) = {my(res = vector(n), t = 2, pit = 1); forprime(p = 3, oo, res[t-1] = p \ pit; if(t > n, return(res) ); t++; if(isprime(t), pit++ ) ) } \\ _David A. Corneth_, Aug 22 2020

%Y Cf. A000040 (primes), A000720 (pi).

%K nonn,easy

%O 2,1

%A _Cino Hilliard_, Oct 14 2005