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.

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, 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, 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

Offset: 0

Author

Harry J. Smith, May 15 2003

Keywords

• nonn

Comments

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.

Examples

			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.
		

References

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

Links

• Eric Weisstein's World of Mathematics, Bertrand's Postulate.

Crossrefs

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