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%I M0854 #50 Dec 26 2021 21:03:01
%S 2,3,7,23,29,157,907,1669,1879,2089,249037,262897,725663,36850999,
%T 173471351,198793279,4827507229,17010526363,83547839407,572945039351,
%U 6269243827111
%N Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.
%C In other words, smallest prime which is at the end of an arithmetic progression of n primes.
%C For the corresponding values of the first term and the common difference, see A113827 and A093364. For the actual arithmetic progressions, see A133277.
%C One may also minimize the common difference: this leads to A033189, A033188 and A113872.
%C One may also specify that the first term is the n-th prime and then minimize the common difference (or, equally, the last term): this leads to A088430 and A113834.
%C One may also ask for n consecutive primes in arithmetic progression: this gives A006560.
%D R. K. Guy, Unsolved Problems in Number Theory, A5.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Jens Kruse Andersen, <a href="http://primerecords.dk/aprecords.htm">Primes in Arithmetic Progression Records</a> [May have candidates for later terms in this sequence.]
%H H. Dubner and H. Nelson, <a href="https://doi.org/10.1090/S0025-5718-97-00875-2">Seven consecutive primes in arithmetic progression</a>, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
%H Ben Green and Terence Tao, <a href="http://arXiv.org/abs/math/0404188">The primes contain arbitrarily long arithmetic progressions</a>, Annals of Mathematics 167 (2008), pp. 481-547. arXiv:math/0404188 [math.NT], 2004-2007.
%H Ben Green and Terence Tao, <a href="http://www.math.ucla.edu/~tao/preprints/Expository/quantitative_AP.dvi">A bound for progressions of length k in the primes</a>
%H Andrew Granville, <a href="http://www.dms.umontreal.ca/~andrew/PDF/PrimePatterns.pdf">Prime number patterns</a>, American Mathematical Monthly 115 (2008), pp. 279-296.
%H A. Moran, P. Pritchard and A. Thyssen, <a href="https://doi.org/10.1090/S0025-5718-1995-1297475-1">Twenty-two primes in arithmetic progression</a>, Math. Comp.64 (1995), no.211, 1337-1339.
%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%F Green & Tao prove that this sequence is infinite, and further a(n) < 2^2^2^2^2^2^2^2^O(n). Granville conjectures that a(n) <= n! + 1 for n >= 3 and give a heuristic suggesting a(n) is around (exp(1-gamma) n/2)^(n/2). - _Charles R Greathouse IV_, Feb 26 2013
%e n, AP, last term
%e 1 2 2
%e 2 2+j 3
%e 3 3+2j 7
%e 4 5+6j 23
%e 5 5+6j 29
%e 6 7+30j 157
%e 7 7+150j 907
%e 8 199+210j 1669
%e 9 199+210j 1879
%e 10 199+210j 2089
%e 11 110437+13860j 249037
%e 12 110437+13860j 262897
%e ..........................
%e a(11)=249037 since 110437,124297,...,235177,249037 is an arithmetic progression of 11 primes ending with 249037 and it is the least number with this property.
%t (* This program will generate the 4 to 12 terms to use a[n_] to generate term 13 or higher, it will have a prolonged run time. *) a[n_] := Module[{i, p, found, j, df, k}, i = 1; While[i++; p = Prime[i]; found = 0; j = 0; While[j++; df = 6*j; (p > ((n - 1)*df)) && (found == 0), found = 1; Do[If[! PrimeQ[p - k*df], found = 0], {k, 1, n - 1}]]; found == 0]; p]; Table[a[i], {i, 4, 12}]
%Y For the associated gaps, see A093364. For the initial terms, see A113827. For the arithmetic progressions, see A133277.
%Y Cf. A006560, A096003, A113830-A113834, A088430.
%K nonn,hard,more,nice
%O 1,1
%A _N. J. A. Sloane_
%E a(11)-a(13) from _Michael Somos_, Mar 14 2004
%E a(14) and corrected version of a(7) from _Hugo Pfoertner_, Apr 27 2004
%E a(15)-a(17) from _Don Reble_, Apr 27 2004
%E a(18)-a(21) from Granville's paper, Jan 26 2006
%E Entry revised by _N. J. A. Sloane_, Jan 26 2006, Oct 17 2007
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