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

1,2

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;
  m
end proc:
map(f, [$1..100]);

Crossrefs

Cf. A000040, A000720, A005115, A373887.

Sequence in context: A245908 A023514 A179751 * A039645 A317993 A048687

Adjacent sequences: A373885 A373886 A373887 * A373889 A373890 A373891

Keyword

nonn

Author

Robert Israel, Aug 11 2024

Status

approved