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 A008407 Minimal difference s(n) between beginning and end of n consecutive large primes (n-tuplet) permitted by divisibility considerations. 29
 0, 2, 6, 8, 12, 16, 20, 26, 30, 32, 36, 42, 48, 50, 56, 60, 66, 70, 76, 80, 84, 90, 94, 100, 110, 114, 120, 126, 130, 136, 140, 146, 152, 156, 158, 162, 168, 176, 182, 186, 188, 196, 200, 210, 212, 216, 226, 236, 240, 246, 252, 254, 264, 270, 272, 278 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Tony Forbes defines a prime k-tuplet (distinguished from a prime k-tuple) to be a maximally possible dense cluster of primes (a prime constellation) which will necessarily involve consecutive primes whereas a prime k-tuple is a prime cluster which may not necessarily be of maximum possible density (in which case the primes are not necessarily consecutive.) a(n) >> n log log n; in particular, for any eps > 0, there is an N such that a(n) > (e^gamma - eps) n log log n for all n > N. Probably N can be chosen as 1; the actual rate of growth is larger. Can a larger growth rate be established? Perhaps a(n) ~ n log n. - Charles R Greathouse IV, Apr 19 2012 Conjecture: (i) The sequence a(n)^(1/n) (n=3,4,...) is strictly decreasing (to the limit 1). (ii) We have 0 < a(n)/n - H_n < (gamma + 2)/(log n) for all n > 4, where H_n denotes the harmonic number 1+1/2+1/3+...+1/n, and gamma refers to the Euler constant 0.5772... [The second inequality has been verified for n = 5, 6, ..., 5000.] - Zhi-Wei Sun, Jun 28 2013. Conjecture: For any integer n > 2, there is 1 < k < n such that 2*n - a(k)- 1 and 2*n - a(k) + 1 are twin primes. Also, every n = 3, 4, ... can be written as p + a(k)/2 with p a prime and k an integer greater than one. - Zhi-Wei Sun, Jun 29-30 2013. The number of configurations that realize this minimal diameter, is A083409(n). - Jeppe Stig Nielsen, Jul 26 2018 REFERENCES R. K. Guy, "Unsolved Problems in Number Theory", lists a number of relevant papers in Section A8. John Leech, "Groups of primes having maximum density", Math. Tables Aids to Comput., 12 (1958) 144-145. LINKS T. D. Noe, Table of n, a(n) for n = 1..672 (from Engelsma's data) Thomas J. Engelsma, Permissible Patterns Tony Forbes, k-tuplets G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923. See final section. A. V. Sutherland, Narrow admissible k-tuples: bounds on H(k), 2013. T. Tao, Bounded gaps between primes, PolyMath Wiki Project, 2013. Eric Weisstein's World of Mathematics, Prime Constellation. FORMULA s(k), k >= 2, is smallest s such that there exist B = {b_1, b_2, ..., b_k} with s = b_k - b_1 and such that for all primes p <= k, not all residues modulo p are represented by B. CROSSREFS Equals A020497 - 1. Cf. A083409. Sequence in context: A285342 A111051 A077561 * A111224 A139718 A173340 Adjacent sequences:  A008404 A008405 A008406 * A008408 A008409 A008410 KEYWORD nonn,nice AUTHOR T. Forbes (anthony.d.forbes(AT)googlemail.com) EXTENSIONS Correction from Pat Weidhaas (weidhaas(AT)wotan.llnl.gov), Jun 15 1997 Edited by Daniel Forgues, Aug 13 2009 a(1)=0 prepended by Max Alekseyev, Aug 14 2015 STATUS approved

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Last modified July 14 00:36 EDT 2020. Contains 335716 sequences. (Running on oeis4.)