cp's OEIS Frontend

A354743 Smallest first term of arithmetic progression of exactly n primes with difference A033188(n).

%I A354743 #15 Jun 08 2022 10:13:13
%S A354743 2,2,3,41,5,7,7,881,3499,199,60858179,147692845283,14933623,
%T A354743 834172298383,894476585908771,1275290173428391,259268961766921,
%U A354743 1027994118833642281
%N A354743 Smallest first term of arithmetic progression of exactly n primes with difference A033188(n).
%C A354743 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.
%C A354743 The word "exactly" requires both i-d and i+n*d to be nonprime.
%C A354743 For the corresponding values of the last term, see A354744.
%C A354743 Without "exactly", we get A033189.
%C A354743 The primes in these arithmetic progressions need not be consecutive.
%C A354743 a(n) != A033189(n) for n = 4, 8, 9, 19 because in these particular cases A113872(n) + A033188(n) is prime.
%C A354743 a(8) = 881 and a(9) = 3499 found by _Michael S. Branicky_ come from A354377.
%C A354743 a(19) > A033189(19) = 1424014323012131633 is not known, it is the smallest first term of an arithmetic progression of exactly 19 primes with a common difference d = 9699690; then a(20) = 1424014323012131633 and a(21) = 28112131522731197609.
%D A354743 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A5, Arithmetic progressions of primes, pp. 25-28.
%H A354743 J. K. Andersen, <a href="http://primerecords.dk/aprecords.htm">Primes in Arithmetic Progression Records</a>.
%H A354743 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeArithmeticProgression.html">Prime Arithmetic Progression</a>.
%H A354743 <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>.
%e A354743 The first few corresponding arithmetic progressions are:
%e A354743 n = 1 and d = 0:   (2);
%e A354743 n = 2 and d = 1:   (2, 3);
%e A354743 n = 3 and d = 2:   (3, 5, 7);
%e A354743 n = 4 and d = 6:   (41, 47, 53, 59);
%e A354743 n = 5 and d = 6:   (5, 11, 17, 23, 29);
%e A354743 n = 6 and d = 30:  (7, 37, 67, 97, 127, 157);
%e A354743 n = 7 and d = 150: (7, 157, 307, 457, 607, 757, 907);
%e A354743 n = 8 and d = 210: (881, 1091,1301, 1511, 1721, 1931, 2141, 2351).
%Y A354743 Cf. A033188, A033189, A113872, A354377, A354744.
%K A354743 nonn,more
%O A354743 1,1
%A A354743 _Bernard Schott_, Jun 05 2022