# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a020497 Showing 1-1 of 1 %I A020497 #49 Oct 26 2023 11:19:11 %S A020497 1,3,7,9,13,17,21,27,31,33,37,43,49,51,57,61,67,71,77,81,85,91,95,101, %T A020497 111,115,121,127,131,137,141,147,153,157,159,163,169,177,183,187,189, %U A020497 197,201,211,213,217,227,237,241,247,253,255,265,271,273,279,283,289,301,305 %N 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. %C A020497 a(n) purportedly gives the least k with A023193(k) = n; that is, this sequence should be the "least inverse" of A023193. %C A020497 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 %C A020497 Tomás Oliveira e Silva (see link) has a table extending to n = 1000. %C A020497 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 %D A020497 R. K. Guy, Unsolved Problems in Number Theory, (2nd edition, Springer, 1994), Section A9. %H A020497 T. D. Noe, Table of n, a(n) for n = 1..672 (from Engelsma's data) %H A020497 Thomas J. Engelsma, Permissible Patterns. %H A020497 T. Forbes, Prime k-tuplets %H A020497 Daniel M. Gordon and Gene Rodemich, Dense admissible sets, Proceedings of ANTS III, LNCS 1423 (1998), pp. 216-225. %H A020497 D. Hensley and I. Richards, Primes in intervals, Acta Arith. 25 (1974), pp. 375-391. %H A020497 H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), pp. 119-134. %H A020497 Tomás Oliveira e Silva, Admissible prime constellations %H A020497 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 A020497 H. Smith, On a generalization of the prime pair problem, Math. Comp., 11 (1957) 249-254. %H A020497 Eric Weisstein's World of Mathematics, k-Tuple Conjecture. %F A020497 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 %Y A020497 Equals A008407 + 1. First differences give A047947. %Y A020497 Cf. A023193 (prime k-tuplet conjectures), A066081 (weaker binary conjectures). %K A020497 nonn,nice %O A020497 1,2 %A A020497 _Robert G. Wilson v_, _Christopher E. Thompson_ %E A020497 Corrected and extended by _David W. Wilson_ # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE