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A020497
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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.
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22
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, (2nd edition, Springer, 1994), Section A9.
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LINKS
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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, Prime k-Tuples Conjecture.
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FORMULA
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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
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CROSSREFS
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Equals A008407 + 1. First differences give A047947.
Cf. A023193 (prime k-tuplet conjectures), A066081 (weaker binary conjectures).
Sequence in context: A130568 A143803 A284894 * A023490 A032375 A089556
Adjacent sequences: A020494 A020495 A020496 * A020498 A020499 A020500
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KEYWORD
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nonn,nice
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AUTHOR
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Robert G. Wilson v, Christopher E. Thompson
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EXTENSIONS
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Corrected and extended by David W. Wilson
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STATUS
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approved
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