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

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, 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, 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

Offset: 1

Author

Robert Israel, Aug 11 2024

Keywords

• nonn

Comments

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.

The first appearance of m in this sequence is at A000720(A005115(m)).

Conjectures: a(n) >= 3 for n >= 13.

Limit_{n -> oo} a(n) = oo.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000720, A005115, A373887.

Programs

Maple

f:= proc(n) local s,i,m,d,j;
  m:= 1;
  s:= ithprime(n);
  for i from n-1 to 1 by -1 do
    d:= s - ithprime(i);
    if s - m*d < 2 then return m fi;
    for j from 2 while isprime(s-j*d) do od;
    m:= max(m, j);
  od;
  m
end proc:
map(f, [$1..100]);