

A023193


Largest number of pairwise coprime numbers that can occur in an interval of length n.


5



1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

Conjecturally, a(n) is the largest number of primes that occurs on an infinite number of intervals of n consecutive integers. The conjecture is apparently due to Dickson; Hardy & Littlewood's Conjecture B concerns only pairs (p, p + 2n).
According to the link at www.opertech.com, a(3159) >= 447 > 446 = pi(3159). The ktuples conjecture then implies that, for an infinitude of n, the interval [n+1, n+3159] includes 447 primes. For these n, pi(n+3159) >= pi(n)+447 > pi(n)+446 = pi(n)+pi(3159), contradicting the conjecture that pi(x+y) <= pi(x)+pi(y).  David W. Wilson, May 23 2005


REFERENCES

Douglas Hensley and Ian Richards, "On the incompatibility of two conjectures concerning primes". Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123127.


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..2330 (based on Engelsma's data)
Thomas J Engelsma, ktuple: Permissible patterns
T. Forbes, Prime ktuplets.
D. Hensley and I. Richards, Primes in intervals, Acta Arith. 25 (1974), pp. 375391.
Ian Richards, On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem, Bulletin of the American Mathematical Society 80:3 (1974), pp. 419438.
Eric Weisstein's World of Mathematics, Prime kTuples Conjecture.


FORMULA

Conjecturally, a(n) = lim sup pi(x+n)pi(x), where pi = A000720. This would follow from the ktuple conjecture.  David W. Wilson, May 23 2005
a(n) = minimum m such that A008407(m) >= n. [From Max Alekseyev, Nov 03 2008]
Richards shows that a(n) > n/log n + kn/log^2 n + o(n/log^2 n), where k = 1 + log 2 = 1.69.... In particular, a(n) > pi(n) for large enough n. Hensley & Richards 1974 cite a result of Montgomery & Vaughan "to appear" that a(n) <= 2pi(n) for n >= 2.  Charles R Greathouse IV, Apr 16 2013


CROSSREFS

Cf. A008407 (minimal difference of first and last prime in a prime ktuplet), A047947 (Schinzel's rhobar), A066081 (weaker binary conjectures), A062571.
Least inverse is A020497.
Sequence in context: A209082 A098424 A098428 * A096605 A189671 A109497
Adjacent sequences: A023190 A023191 A023192 * A023194 A023195 A023196


KEYWORD

nonn,nice


AUTHOR

David W. Wilson


STATUS

approved



