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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.

%I A020497 #50 Feb 16 2025 08:32:33
%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, <a href="/A020497/b020497.txt">Table of n, a(n) for n = 1..672</a> (from Engelsma's data)
%H A020497 Thomas J. Engelsma, <a href="http://www.opertech.com/primes/k-tuples.html">Permissible Patterns</a>.
%H A020497 T. Forbes, <a href="http://anthony.d.forbes.googlepages.com/adf.htm">Prime k-tuplets</a>
%H A020497 Daniel M. Gordon and Gene Rodemich, <a href="https://dmgordon.org/papers/ants.pdf">Dense admissible sets</a>, Proceedings of ANTS III, LNCS 1423 (1998), pp. 216-225.
%H A020497 D. Hensley and I. Richards, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa25/aa2548.pdf">Primes in intervals</a>, Acta Arith. 25 (1974), pp. 375-391.
%H A020497 H. L. Montgomery and R. C. Vaughan, <a href="http://dx.doi.org/10.1112/S0025579300004708">The large sieve</a>, Mathematika 20 (1973), pp. 119-134.
%H A020497 Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/apc.html">Admissible prime constellations</a>
%H A020497 Ian Richards, <a href="http://projecteuclid.org/euclid.bams/1183535510">On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem</a>, Bulletin of the American Mathematical Society 80:3 (1974), pp. 419-438.
%H A020497 H. Smith, <a href="http://dx.doi.org/10.1090/S0025-5718-1957-0094314-8">On a generalization of the prime pair problem</a>, Math. Comp., 11 (1957) 249-254.
%H A020497 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>.
%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_