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A354376 Smallest prime which is at the end of an arithmetic progression of exactly n primes.

%I A354376 #53 May 29 2022 08:08:45
%S A354376 2,3,7,43,29,157,907,2351,5179,2089,375607,262897,725663,36850999,
%T A354376 173471351,198793279,4827507229,17010526363,83547839407,572945039351,
%U A354376 6269243827111
%N A354376 Smallest prime which is at the end of an arithmetic progression of exactly n primes.
%C A354376 Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.
%C A354376 The word "exactly" requires both i-d and i+n*d to be nonprime; without "exactly", we get A005115.
%C A354376 For the corresponding values of the first term, and the common difference, see A354377 and A354484. For the actual arithmetic progressions, see A354485.
%C A354376 The primes in these arithmetic progressions need not be consecutive. (The smallest prime at the start of a run of exactly n consecutive primes in arithmetic progression is A006560(n).)
%C A354376 a(n) != A005115(n), because A005115(n) + A093364(n) is prime for n = 4, 8, 9, 11. - _Michael S. Branicky_, May 24 2022
%D A354376 R. K. Guy, Unsolved Problems in Number Theory, A5, Arithmetic progressions of primes.
%e A354376 The arithmetic progression (5, 11, 17, 23) with common difference 6 contains 4 primes, but 29 = 23+6 is also prime, hence a(4) != 23.
%e A354376 The arithmetic progression (7, 19, 31, 43) with common difference 12 also contains 4 primes, and 7-12 < 0 and 43+12 = 55 is composite; moreover this arithmetic progression is the smallest such progression with exactly 4 primes, hence a(4) = 43.
%o A354376 (Python)
%o A354376 from sympy import isprime, nextprime
%o A354376 def a(n):
%o A354376     if n < 3: return [2, 3][n-1]
%o A354376     p = 2
%o A354376     while True:
%o A354376         for d in range(2, (p-3)//(n-1)+1, 2):
%o A354376             if isprime(p+d) or isprime(p-n*d): continue
%o A354376             if all(isprime(p-j*d) for j in range(1, n)): return p
%o A354376         p = nextprime(p)
%o A354376 print([a(n) for n in range(1, 11)]) # _Michael S. Branicky_, May 24 2022
%Y A354376 Cf. A005115, A006560, A093364, A354377, A354484, A354485.
%K A354376 nonn,more
%O A354376 1,1
%A A354376 _Bernard Schott_, May 24 2022
%E A354376 a(4) corrected and a(8)-a(13) from _Michael S. Branicky_, May 24 2022
%E A354376 a(14)-a(21) derived using A005115 and A093364 by _Michael S. Branicky_, May 24 2022