A373888 - OEIS

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

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OFFSET

1,2

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.

LINKS

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

EXAMPLE

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.

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;

end proc:

map(f, [$1..100]);