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.

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, 6, 30, 30, 30, 24, 36, 48, 24, 30, 12, 18, 42, 6, 54, 54, 42, 48, 60, 30, 42, 30, 66, 42, 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

Offset: 2

Author

Robert Israel, Aug 13 2024

Keywords

• nonn
• look

Comments

a(n) is the smallest common difference in an arithmetic progression of A373888(n) primes ending in prime(n).

a(n) is divisible by all primes < min(A373888(n) + 1, prime(n) - (A373888(n)-1)*a(n)).

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A000040, A373888.

Programs

Maple

f:= proc(n) local s, i, m, dd, 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 dd fi;
    for j from 2 while isprime(s-j*d) do od;
    if j > m then m:= j; dd:= d fi;
  od;
  dd
end proc:
map(f, [$2..100]);