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

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

Offset: 1

Author

T. Forbes (anthony.d.forbes(AT)googlemail.com)

Keywords

• nonn
• nice

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

Engelsma points out that the values he lists past a(342)=2328 are only the "best known" values, and are not confirmed as minimal. - Brian Kehrig, Mar 21 2025

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..342 (shortened to 342 terms by Brian Kehrig)
• Thomas J. Engelsma, Permissible Patterns
• Tony Forbes and Norman Luhn, k-tuplets
• Daniel A. Goldston, Apoorva Panidapu, and Jordan Schettler, Explicit Calculations for Sono's Multidimensional Sieve of E2-Numbers, arXiv:2208.13931 [math.NT], 2022. See H(n) in Table 1 p. 2.
• 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.
• Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.
• 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.

Crossrefs

Equals A020497 - 1.

Cf. A083409.

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.

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