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%I #35 Jun 05 2022 03:40:49
%S 0,1,2,12,6,30,150,210,210,210,30030,13860,60060,420420,4144140,
%T 9699690,87297210,717777060,4180566390,18846497670,26004868890
%N Common differences associated with the arithmetic progressions of primes in A354376.
%C 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) = common difference d.
%C The word "exactly" requires both i-d and i+n*d to be nonprime; without "exactly", we get A093364.
%C For the corresponding values of the first term and the last term, see respectively A354377 and A354376. For the actual arithmetic progressions, see A354485.
%C 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).)
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A5, Arithmetic progressions of primes, pp. 25-28.
%F a(1) = 0, then for n > 1, a(n) = (A354376(n) - A354377(n)) / (n-1).
%e The first few corresponding arithmetic progressions are:
%e d = 0: (2);
%e d = 1: (2, 3);
%e d = 2: (3, 5, 7);
%e d = 12: (7, 19, 31, 43);
%e d = 6: (5, 11, 17, 23, 29);
%e d = 30: (7, 37, 67, 97, 127, 157);
%e d = 150: (7, 157, 307, 457, 607, 757, 907).
%Y Cf. A006560, A093364, A354376, A354377, A354485.
%K nonn,more
%O 1,3
%A _Bernard Schott_, May 28 2022
%E a(7)-a(21) via A354376, A354377 from _Michael S. Branicky_, May 28 2022
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