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A375386 a(n) is the common difference in the longest arithmetic progression of primes ending in prime(n). If there is more than one such arithmetic progression, the smallest difference is chosen.

%I A375386 #8 Aug 14 2024 08:33:53
%S A375386 1,2,2,4,2,6,6,6,6,12,6,12,12,18,12,6,24,24,12,6,6,12,18,18,30,30,18,
%T A375386 6,30,30,30,24,36,48,24,30,12,18,42,6,54,54,42,48,60,30,42,30,66,42,
%U A375386 66,30,60,30,12,6,30,48,84,60,60,78,60,102,60,60,30,78,36,60,90,18,90,6,72,96,30,54
%N A375386 a(n) is the common difference in the longest arithmetic progression of primes ending in prime(n).  If there is more than one such arithmetic progression, the smallest difference is chosen.
%C A375386 a(n) is the smallest common difference in an arithmetic progression of A373888(n) primes ending in prime(n).
%C A375386 a(n) is divisible by all primes < min(A373888(n) + 1, prime(n) - (A373888(n)-1)*a(n)).
%H A375386 Robert Israel, <a href="/A375386/b375386.txt">Table of n, a(n) for n = 2..10000</a>
%e A375386 a(4) = 2 because the 4th prime is 7 and the arithmetic progression of 3 primes ending in 7, namely 3, 5, 7, has common difference 2.
%p A375386 f:= proc(n) local s, i, m, dd, d, j;
%p A375386   m:= 1;
%p A375386   s:= ithprime(n);
%p A375386   for i from n-1 to 1 by -1 do
%p A375386     d:= s - ithprime(i);
%p A375386     if s - m*d < 2 then return dd fi;
%p A375386     for j from 2 while isprime(s-j*d) do od;
%p A375386     if j > m then m:= j; dd:= d fi;
%p A375386   od;
%p A375386   dd
%p A375386 end proc:
%p A375386 map(f, [$2..100]);
%Y A375386 Cf. A000040, A373888.
%K A375386 nonn,look
%O A375386 2,2
%A A375386 _Robert Israel_, Aug 13 2024