A316791 - OEIS

%I #16 Mar 08 2023 08:23:58

%S 3,5,29,503,137,109,1063,1931,521,7951,1949,1667,5743,2969,1321,15817,

%T 9547,28349,45433,20807,15679,113837,43793,19603,40283,25469,40637,

%U 156151,79697,34057,282487,134507,552401,770663,31393,188021,480203,461707,281429,1078241,265619,637937

%N a(n) is the least prime p such that the second forward difference of three consecutive primes p, q and r is n = (p - 2q + r)/2.

%C Inspired by A295973.

%C Except for the first three primes {2, 3, 5}, all sfds are even.

%C The only other sfd which is not covered by this sequence is when the primes are {2, 3, 5} which results in an sfd of 1.

%C Except for an sfd of 0 or 1, all values of sfd occur infinitely often.

%C As an example, sfd=2 for p = 5, 11, 17, 19, 41, 43, 79, 83, 101, 107, 127, 163, ..., .

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ForwardDifference.html">Forward Difference</a>.

%e a(0) = 3 since the three consecutive primes {3, 5, 7} have an sfd of 0;

%e a(1) = 5 since the three consecutive primes {5, 7, 11} have an sfd of 2;

%e a(2) = 29 since the three consecutive primes {29, 31, 37} have an sfd of 4;

%e a(3) = 503 since the three consecutive primes {503, 509, 521} have an sfd of 6;

%e a(4) = 137 since the three consecutive primes {137, 139, 149} have an sfd of 8; etc.

%t p = 2; q = 3; r = 5; t[_] := 0; While[p < 1100000, d = p - 2q + r; If[ t[d] == 0, t[d] = p]; p = q; q = r; r = NextPrime@ r]; Array[ t[2#] &, 42, 0]

%o (PARI) a(n) = my(p=2, q=3); while ((p - 2*q + nextprime(q+1))/2 != n, p=q; q=nextprime(q+1)); p; \\ _Michel Marcus_, Mar 08 2023

%Y Cf. A000040, A000230, A036263, A137501, A295746, A295973, A316792.

%K nonn

%O 0,1

%A _Edward Bernstein_ and _Robert G. Wilson v_, Jul 14 2018