

A088430


a(n) = the least positive d such that for p=prime(n), the numbers p+0d, p+1d, p+2d, ..., p+(p1)d are all primes.


10




OFFSET

1,2


COMMENTS

Problem discussed by Russell E. Rierson: starting with given p, find the least d such that the arithmetic progression p,p+d,p+2d,... contains only primes. Obviously, the maximum number of prime terms is p and to reach that maximum, d must be a multiple of all smaller primes. For example, a(5) is a multiple of 2*3*5*7.
There can be other maximumlength prime progressions starting at p, with larger d. (Zak Seidov found d=4911773580 for p=11.)


LINKS

Table of n, a(n) for n=1..8.
Jens Kruse Andersen, Smallest APk with minimal start
Phil Carmody, a(7)
Andrew Granville, Prime number patterns
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, arXiv:math/0404188 [math.NT], 20042007. [Background]
Russell E. Rierson, Question About Prime Numbers.
Zak Seidov, Question About Prime Numbers.


FORMULA

a(n) = A231017(n)  prime(n).  Jonathan Sondow, Nov 08 2013
a(n) = A061558(prime(n)).  Jens Kruse Andersen, Jun 30 2014
a(n) = A002110(n1) * A231018(n).  Jeppe Stig Nielsen, Mar 16 2016


EXAMPLE

n AP Last term

1 2+i 3
2 3+2*i 7
3 5+6*i 29
4 7+150*i 907
5 11+1536160080*i 15361600811
6 13+9918821194590*i 119025854335093
7 17+341976204789992332560*i 5471619276639877320977
8 19+2166703103992332274919550*i 39000655871861980948551919


CROSSREFS

See A113834 for last term in the progression, and A231017 for the 2nd term.
Cf. A061558, A231018, A002110.
Sequence in context: A024397 A015173 A122570 * A246958 A219761 A051240
Adjacent sequences: A088427 A088428 A088429 * A088431 A088432 A088433


KEYWORD

more,nonn


AUTHOR

Zak Seidov, Sep 30 2003


EXTENSIONS

Edited by Don Reble, Oct 04 2003
a(7) was found by Phil Carmody.  Don Reble, Nov 23 2003
Entry revised by N. J. A. Sloane, Jan 25 2006
a(8) found by Wojciech Izykowski.  Jens Kruse Andersen, Jun 30 2014


STATUS

approved



