A084142 Positive numbers k such that the number of primes between k and 2*k is different from the number of primes between m and 2*m for every number m != k.

1, 120, 216, 300, 531, 714, 804, 999, 1344, 1356, 1395, 1680, 1764, 1770, 1959, 2046, 2121, 2325, 2484, 2511, 2760, 2826, 3150, 3285, 3396, 3744, 4044, 4116, 4146, 4314, 4710, 4839, 5046, 5070, 5136, 5250, 5586, 5970, 6411, 6459, 6501, 6504, 6846, 7275

Offset: 1

Author

Harry J. Smith, May 15 2003

Keywords

• nonn

Comments

The number of primes between k and 2*k is unique because no other number m > 0 has the same of primes between m and 2m, exclusively. k is the value of A060756(j) or A084139(j) when A084138(j) = 1. Question: Is this sequence infinitely long?

If k > 1 is a term then A060715(k-1) < A060715(k) < A060715(k+1). Consequently, (2*k-1, 2*k+1) is a twin prime pair, so 3|k. Additionally, it can be shown that k-1..k+3 are all composite numbers. Moreover, if k is even, then k-4..k+4 are all composite numbers. - Jon E. Schoenfield, Oct 08 2023

Examples

			120 is a term because there are 22 primes between 120 and 240 and no other number m > 0 has 22 primes between m and 2*m.
		

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, A084138, A084139, A084140, A084141.

Extensions

Name edited by Jon E. Schoenfield, Oct 08 2023