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A008407
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Minimal difference s(n) between beginning and end of n consecutive large primes (n-tuplet) permitted by divisibility considerations.
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29
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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
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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T. Forbes (anthony.d.forbes(AT)googlemail.com)
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EXTENSIONS
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Correction from Pat Weidhaas (weidhaas(AT)wotan.llnl.gov), Jun 15 1997
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STATUS
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approved
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