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A088430 a(n) = the least positive d such that for p=prime(n), the numbers p+0d, p+1d, p+2d, ..., p+(p-1)d are all primes. 10
1, 2, 6, 150, 1536160080, 9918821194590, 341976204789992332560, 2166703103992332274919550 (list; graph; refs; listen; history; text; internal format)



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 maximum-length prime progressions starting at p, with larger d. (Zak Seidov found d=4911773580 for p=11.)


Table of n, a(n) for n=1..8.

Jens Kruse Andersen, Smallest AP-k 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], 2004-2007. [Background]

Russell E. Rierson, Question About Prime Numbers.

Zak Seidov, Question About Prime Numbers.


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(n-1) * A231018(n). - Jeppe Stig Nielsen, Mar 16 2016


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


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




Zak Seidov, Sep 30 2003


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



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Last modified November 20 12:23 EST 2017. Contains 294971 sequences.