A349642 - OEIS

%I #10 Dec 05 2021 14:32:08

%S 2,2,2,17,347,2903,15373,128981,19641263,245333213,245333213,

%T 27797667517,68439250465123,68439250465123

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

%C 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.

%C 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.

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

%e 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.

%e 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.

%Y From n = 3, second row of A349644.

%Y Cf. A001223, A006560, A036263, A348927.

%K nonn,more

%O 0,1

%A _Pontus von Brömssen_, Nov 23 2021

%E a(12)-a(13) from _Martin Ehrenstein_, Dec 05 2021