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A008407 Minimal difference s(n) between beginning and end of n consecutive large primes (n-tuplet) permitted by divisibility considerations. 23
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

2,1

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(1) would be 0 (for a prime 1-tuplet.)

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.

REFERENCES

R. K. Guy, "Unsolved Problems in Number Theory", lists a number of relevant papers in Section A8.

G. H. Hardy and J.E. Littlewood, "Partitio Numerorum III", Acta Math. 44 (1922) 1-70, see final section.

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 = 2..672 (from Engelsma's data)

Thomas J. Engelsma, Permissible Patterns

Tony Forbes, k-tuplets

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.

Sequence in context: A189400 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 weidhaas(AT)wotan.llnl.gov (Pat Weidhaas) Jun 15 1997.

Edited by Daniel Forgues, Aug 13 2009

STATUS

approved

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Last modified October 22 04:03 EDT 2014. Contains 248388 sequences.