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A373888 a(n) is the length of the longest arithmetic progression of primes ending with prime(n).

%I A373888 #27 Aug 13 2024 09:10:26
%S A373888 1,2,2,3,3,2,3,3,4,5,3,2,4,4,3,5,4,3,3,3,3,4,4,3,4,4,4,4,3,4,5,5,3,4,
%T A373888 4,4,6,4,4,5,3,4,4,4,5,4,3,4,5,4,4,4,4,5,6,4,4,5,3,4,5,5,4,6,4,4,4,3,
%U A373888 4,4,6,4,4,5,3,4,5,5,4,4,4,5,4,4,4,5,5,4,4,6,4,5,4,4,3,4,6,5,4
%N A373888 a(n) is the length of the longest arithmetic progression of primes ending with prime(n).
%C A373888 a(n) is the greatest k such that there exists d > 0 such that A000040(n) - j*d is prime for j = 0 .. k-1.
%C A373888 The first appearance of m in this sequence is at A000720(A005115(m)).
%C A373888 Conjectures: a(n) >= 3 for n >= 13.
%C A373888 Limit_{n -> oo} a(n) = oo.
%H A373888 Robert Israel, <a href="/A373888/b373888.txt">Table of n, a(n) for n = 1..10000</a>
%e A373888 a(4) = 3 because the 4th prime is 7 and there is an arithmetic progression of 3 primes ending in 7, namely 3, 5, 7, and no such arithmetic progression of 4 primes.
%p A373888 f:= proc(n) local s,i,m,d,j;
%p A373888   m:= 1;
%p A373888   s:= ithprime(n);
%p A373888   for i from n-1 to 1 by -1 do
%p A373888     d:= s - ithprime(i);
%p A373888     if s - m*d < 2 then return m fi;
%p A373888     for j from 2 while isprime(s-j*d) do od;
%p A373888     m:= max(m, j);
%p A373888   od;
%p A373888   m
%p A373888 end proc:
%p A373888 map(f, [$1..100]);
%Y A373888 Cf. A000040, A000720, A005115, A373887.
%K A373888 nonn
%O A373888 1,2
%A A373888 _Robert Israel_, Aug 11 2024