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A084138 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.

%I A084138 #13 Feb 16 2025 08:32:49
%S A084138 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 A084138 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 A084138 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 A084138 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 A084138 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 A084138 P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 140.
%H A084138 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate.</a>
%e A084138 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 A084138 Cf. A060715, A060756, A084139, A084140, A084141, A084142.
%K A084138 nonn
%O A084138 0,2
%A A084138 _Harry J. Smith_, May 15 2003