A349642 Smallest prime such that the next n prime gaps are in arithmetic progression.

2, 2, 2, 17, 347, 2903, 15373, 128981, 19641263, 245333213, 245333213, 27797667517, 68439250465123, 68439250465123

Offset: 0

Author

Pontus von Brömssen, Nov 23 2021

Keywords

• nonn
• more

Comments

Equivalently, a(n) is the smallest prime p = prime(k) such that there is a polynomial f of degree at most 2 such that f(j) = prime(j) for k <= j <= k + n.

Any sequence of at most 2 terms is considered to be a degenerate arithmetic progression, so a(n) = 2 (the smallest prime) for n <= 2.

a(n) is the smallest prime p = prime(k) such that A036263(k) = A036263(k+1) = ... = A036263(k+n-2).

Examples

			The three prime gaps following the prime 17 are 2, 4, and 6, which are in arithmetic progression. This is not true for any smaller prime, so a(3) = 17.
The eight prime gaps following the prime 19641263 are 20, 18, 16, 14, 12, 10, 8, and 6, which are in arithmetic progression. This is not true for any smaller prime, so a(8) = 19641263.
		

Crossrefs

From n = 3, second row of A349644.

Cf. A001223, A006560, A036263, A348927.

Extensions

a(12)-a(13) from Martin Ehrenstein, Dec 05 2021