A084138 - OEIS

%I #12 Jul 29 2017 19:26:11

%S 1,3,4,4,7,3,5,6,2,9,6,2,5,10,7,8,5,3,9,10,6,4,1,8,6,5,5,9,11,10,6,6,

%T 10,8,5,6,1,3,8,9,9,5,18,16,5,7,3,1,3,12,5,3,3,3,9,8,16,7,5,8,15,10,4,

%U 2,8,7,10,13,17,5,8,7,9,10,3,5,3,6,6,1,6,8,3,3,10,15,14,16,7,10,14,5,5,3,8

%N a(n) is the number of times n is in sequence A060715, i.e., there are exactly a(n) cases where there are exactly n primes between m and 2m, exclusively, for m > 0.

%C This calculation relies on the fact that Pi(2*m) - Pi(m) > m/(3*log(m)) for m >= 5. It can be shown that a(n) is never zero, i.e., every nonnegative integer is in sequence A060715.

%D P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 140.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate.</a>

%e a(22)=1 because there are 22 primes between 120 and 240 (namely, prime numbers p(31)=127 through p(52)=239), and in no other case are there exactly 22 primes between m and 2m exclusively.

%Y Cf. A060715, A060756, A084139, A084140, A084141, A084142.

%K nonn

%O 0,2

%A _Harry J. Smith_, May 15 2003