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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. 11
 1, 2, 6, 150, 1536160080, 9918821194590, 341976204789992332560, 2166703103992332274919550 (list; graph; refs; listen; history; text; internal format)
 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 maximum-length prime progressions starting at p, with larger d. (Zak Seidov found d=4911773580 for p=11.) LINKS Jens Kruse Andersen, Smallest AP-k with minimal start Phil Carmody, a(7), NMBRTHRY Nov 2001 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] P. Ribenboium, Les records des nombres premiers, Sem. Phil. Mathem. (8) (1987) 1-25. P. Ribenboim, Prime number records, Coll. Math. J. 25 (4) (1994) 280-290. P. Ribenboim, Euler's Famous prime generating polynomial and the class number of imaginary quadratic fields, (2000) p 91-111 Russell E. Rierson, Question About Prime Numbers. Zak Seidov, Question About Prime Numbers. Zak Seidov and others, Russell E. Rierson's Question About Prime Numbers, digest of 5 messages in primenumbers Yahoo group, Sep 29 - Oct 1, 2003. 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(n-1) * 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 MATHEMATICA A088430[n_] := Module[{p, m, d},    p = Prime[n]; m = Product[Prime[i], {i, 1, n - 1}];    d = m;    While[! AllTrue[Table[p + i*d, {i, 1, p - 1}], PrimeQ], d = d + m];    Return[d];    ]; Table[A088430[n], {n, 1, 8}] (* Robert Price, Mar 27 2019 *) CROSSREFS See A113834 for last term in the progression, and A231017 for the 2nd term. Cf. A061558, A231018, A002110. Sequence in context: A099185 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

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Last modified April 4 19:22 EDT 2020. Contains 333229 sequences. (Running on oeis4.)