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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.

%I A088430 #44 Apr 18 2025 08:55:22
%S A088430 1,2,6,150,1536160080,9918821194590,341976204789992332560,
%T A088430 2166703103992332274919550
%N 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.
%C A088430 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.
%C A088430 There can be other maximum-length prime progressions starting at p, with larger d. (Zak Seidov found d=4911773580 for p=11.)
%D A088430 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 139-140.
%H A088430 Jens Kruse Andersen, <a href="http://primerecords.dk/aprecords.htm#minimalstart">Smallest AP-k with minimal start</a>
%H A088430 Phil Carmody, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;8b742242.0111">a(7)</a>, NMBRTHRY Nov 2001
%H A088430 Andrew Granville, <a href="http://www.dms.umontreal.ca/~andrew/PDF/PrimePatterns.pdf">Prime number patterns</a>
%H A088430 Ben Green and Terence Tao, <a href="https://arxiv.org/abs/math/0404188">The primes contain arbitrarily long arithmetic progressions</a>, arXiv:math/0404188 [math.NT], 2004-2007. [Background]
%H A088430 Paulo Ribenboim, <a href="https://www.numdam.org/item/SPHM_1987___8_A1_0">Les records des nombres premiers</a>, Séminaire de Philosophie et Mathématiques (8) (1987) 1-25.
%H A088430 Paulo Ribenboim, <a href="https://www.researchgate.net/profile/Paulo_Ribenboim/publication/237778627_Prime_Number_Records/links/54fa1e5d0cf23e66f0311635.pdf">Prime number records</a>, Coll. Math. J. 25 (4) (1994) 280-290.
%H A088430 Paulo Ribenboim, <a href="http://dx.doi.org/10.1007/0-387-22754-7_5">Euler's Famous prime generating polynomial and the class number of imaginary quadratic fields</a>, (2000) p 91-111
%H A088430 Russell E. Rierson, <a href="http://mathforum.org/kb/thread.jspa?forumID=13&amp;threadID=30306&amp;messageID=95983#95983">Question About Prime Numbers</a>.
%H A088430 Zak Seidov, <a href="http://groups.yahoo.com/group/primenumbers/message/13656">Question About Prime Numbers</a>.
%H A088430 Zak Seidov and others, <a href="/A088430/a088430.txt">Russell E. Rierson's Question About Prime Numbers</a>, digest of 5 messages in primenumbers Yahoo group, Sep 29 - Oct 1, 2003.
%F A088430 a(n) = A231017(n) - prime(n). - _Jonathan Sondow_, Nov 08 2013
%F A088430 a(n) = A061558(prime(n)). - _Jens Kruse Andersen_, Jun 30 2014
%F A088430 a(n) = A002110(n-1) * A231018(n). - _Jeppe Stig Nielsen_, Mar 16 2016
%e A088430 n AP Last term
%e A088430 --------------
%e A088430 1 2+i 3
%e A088430 2 3+2*i 7
%e A088430 3 5+6*i 29
%e A088430 4 7+150*i 907
%e A088430 5 11+1536160080*i 15361600811
%e A088430 6 13+9918821194590*i 119025854335093
%e A088430 7 17+341976204789992332560*i 5471619276639877320977
%e A088430 8 19+2166703103992332274919550*i 39000655871861980948551919
%t A088430 A088430[n_] := Module[{p, m, d},
%t A088430    p = Prime[n]; m = Product[Prime[i], {i, 1, n - 1}];
%t A088430    d = m;
%t A088430    While[! AllTrue[Table[p + i*d, {i, 1, p - 1}], PrimeQ], d = d + m];
%t A088430    Return[d];
%t A088430    ];
%t A088430 Table[A088430[n], {n, 1, 8}] (* _Robert Price_, Mar 27 2019 *)
%Y A088430 See A113834 for last term in the progression, and A231017 for the 2nd term.
%Y A088430 Cf. A061558, A231018, A002110.
%K A088430 more,nonn
%O A088430 1,2
%A A088430 _Zak Seidov_, Sep 30 2003
%E A088430 Edited by _Don Reble_, Oct 04 2003
%E A088430 a(7) was found by Phil Carmody. - _Don Reble_, Nov 23 2003
%E A088430 Entry revised by _N. J. A. Sloane_, Jan 25 2006
%E A088430 a(8) found by Wojciech Izykowski. - _Jens Kruse Andersen_, Jun 30 2014