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 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.
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
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