Search: id:A005115
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%I A005115 M0854
%S A005115 2,3,7,23,29,157,907,1669,1879,2089,249037,262897,725663,36850999,
%T A005115 173471351,198793279,4827507229,17010526363,83547839407,
%U A005115 572945039351,6269243827111
%N A005115 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 A005115 In other words, smallest prime which is at the end of an arithmetic progression 
               of n primes.
%C A005115 For the corresponding values of the first term and the common difference 
               see A113827 and A093364. For the actual arithmetic progressions see 
               A133277.
%C A005115 One may also minimize the common difference: this leads to A033189, A033188 
               and A113872.
%C A005115 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 A005115 One may also ask for n consecutive primes in arithmetic progression: 
               this gives A006560.
%D A005115 H. Dubner and H. Nelson, Seven consecutive primes in arithmetic progression, 
               Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
%D A005115 R. K. Guy, Unsolved Problems in Number Theory, A5.
%D A005115 A. Moran, P. Pritchard and A. Thyssen, Twenty-two primes in arithmetic 
               progression, Math. Comp.64 (1995), no.211, 1337-1339.
%D A005115 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A005115 Jens Kruse Andersen, <a href="http://users.cybercity.dk/~dsl522332/math/
               aprecords.htm">Primes in Arithmetic Progression Records</a> [May 
               have candidates for later terms in this sequence.]
%H A005115 Ben Green and Terence Tao, <a href="http://arXiv.org/abs/math/0404188">
               The primes contain arbitrarily long arithmetic progressions</a>
%H A005115 Andrew Granville, <a href="http://www.dms.umontreal.ca/~andrew/PDF/PrimePatterns.pdf">
               Prime number patterns</a>
%H A005115 <a href="Sindx_Pri.html#primes_AP">Index entries for sequences related 
               to primes in arithmetic progressions</a>
%e A005115 n, AP, last term
%e A005115 1 2 2
%e A005115 2 2+j 3
%e A005115 3 3+2j 7
%e A005115 4 5+6j 23
%e A005115 5 5+6j 29
%e A005115 6 7+30j 157
%e A005115 7 7+150j 907
%e A005115 8 199+210j 1669
%e A005115 9 199+210j 1879
%e A005115 10 199+210j 2089
%e A005115 11 110437+13860j 249037
%e A005115 12 110437+13860j 262897
%e A005115 ..........................
%e A005115 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.
%Y A005115 For the associated gaps see A093364, for the initial terms see A113827. 
               Cf. A006560, A096003.
%Y A005115 Cf. A113830-A113834, A088430.
%Y A005115 Sequence in context: A156615 A158054 A134412 this_sequence A113872 A120302 
               A093363
%Y A005115 Adjacent sequences: A005112 A005113 A005114 this_sequence A005116 A005117 
               A005118
%K A005115 nonn,nice,hard,more
%O A005115 1,1
%A A005115 N. J. A. Sloane (njas(AT)research.att.com).
%E A005115 a(11)-a(13) from Michael Somos, Mar 14 2004.
%E A005115 a(14) and corrected version of a(7) from Hugo Pfoertner (hugo(AT)pfoertner.org), 
               Apr 27 2004
%E A005115 a(15)-a(17) from Don Reble (djr(AT)nk.ca), Apr 27 2004
%E A005115 a(18)-a(21) from Granville's paper, Jan 26, 2006
%E A005115 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Jan 26 2006, 
               Oct 17 2007

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