A020497 Conjecturally, this is the minimal y such that n primes occur infinitely often among (x+1, ..., x+y), that is, pi(x+y) - pi(x) >= n for infinitely many x.

1, 3, 7, 9, 13, 17, 21, 27, 31, 33, 37, 43, 49, 51, 57, 61, 67, 71, 77, 81, 85, 91, 95, 101, 111, 115, 121, 127, 131, 137, 141, 147, 153, 157, 159, 163, 169, 177, 183, 187, 189, 197, 201, 211, 213, 217, 227, 237, 241, 247, 253, 255, 265, 271, 273, 279, 283, 289, 301, 305

Offset: 1

Author

Robert G. Wilson v, Christopher E. Thompson

Keywords

• nonn
• nice

Comments

a(n) purportedly gives the least k with A023193(k) = n; that is, this sequence should be the "least inverse" of A023193.

My web page extends the sequence to rho(305)=2047 and also gives a super-dense occurrence at rho(592)=4333 when pi(4333)=591 - the first known occurrence. - Thomas J Engelsma (tom(AT)opertech.com), Feb 16 2004

Tomás Oliveira e Silva (see link) has a table extending to n = 1000.

The minimal y such that there are n elements of {1, ..., y} with fewer than p distinct elements mod p for all prime p. - Charles R Greathouse IV, Jun 13 2013

References

• R. K. Guy, Unsolved Problems in Number Theory, (2nd edition, Springer, 1994), Section A9.

Links

• T. D. Noe, Table of n, a(n) for n = 1..672 (from Engelsma's data)
• Thomas J. Engelsma, Permissible Patterns.
• T. Forbes, Prime k-tuplets
• Daniel M. Gordon and Gene Rodemich, Dense admissible sets, Proceedings of ANTS III, LNCS 1423 (1998), pp. 216-225.
• D. Hensley and I. Richards, Primes in intervals, Acta Arith. 25 (1974), pp. 375-391.
• H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), pp. 119-134.
• Tomás Oliveira e Silva, Admissible prime constellations
• Ian Richards, On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem, Bulletin of the American Mathematical Society 80:3 (1974), pp. 419-438.
• H. Smith, On a generalization of the prime pair problem, Math. Comp., 11 (1957) 249-254.
• Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Crossrefs

Equals A008407 + 1. First differences give A047947.

Cf. A023193 (prime k-tuplet conjectures), A066081 (weaker binary conjectures).

Formula

Prime(floor((n+1)/2)) <= a(n) < prime(n) for large n. See Hensley & Richards and Montgomery & Vaughan. - Charles R Greathouse IV, Jun 18 2013

Extensions

Corrected and extended by David W. Wilson