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%I #11 Jun 08 2022 10:13:05
%S 2,3,7,59,29,157,907,2351,5179,2089,60881279,147692870693,15293983,
%T 834172688773,894476586329191,1275290173878841,259268969935081,
%U 1027994118842320951
%N Last term of arithmetic progression of exactly n primes with difference A033188(n) and first term = A354743(n).
%C Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the first one which minimizes the common difference d; then a(n) = i+(n-1)d.
%C The word "exactly" requires both i-d and i+n*d to be nonprime.
%C Without "exactly", we get A113872.
%C The primes in these arithmetic progressions need not be consecutive.
%C a(n) != 113872(n) for n = 4, 8, 9, 19 because in these particular cases A113872(n) + A033188(n) is prime.
%C a(8) = 2351 and a(9) = 5179, found by _Michael S. Branicky_ come from A354376.
%C a(19) > A113872(19) = 1424014323186726053 is not known, it is the last term of the arithmetic progression of exactly 19 primes with a common difference d = 9699690 and first term = A354743(19); then a(20) = 1424014323196425743 and a(21) = 28112131522925191409.
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A5, Arithmetic progressions of primes, pp. 25-28.
%H J. K. Andersen, <a href="http://primerecords.dk/aprecords.htm">Primes in Arithmetic Progression Records</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeArithmeticProgression.html">Prime Arithmetic Progression</a>.
%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>.
%e The first few corresponding arithmetic progressions are:
%e n = 1 and d = 0: (2);
%e n = 2 and d = 1: (2, 3);
%e n = 3 and d = 2: (3, 5, 7);
%e n = 4 and d = 6: (41, 47, 53, 59);
%e n = 5 and d = 6: (5, 11, 17, 23, 29);
%e n = 6 and d = 30: (7, 37, 67, 97, 127, 157);
%e n = 7 and d = 150: (7, 157, 307, 457, 607, 757, 907);
%e n = 8 and d = 210: (881, 1091,1301, 1511, 1721, 1931, 2141, 2351).
%Y Cf. A033188, A033189, A113872, A354377, A354743.
%K nonn,more
%O 1,1
%A _Bernard Schott_, Jun 05 2022
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