A316792 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.

3, 7, 23, 1531, 139, 113, 523, 1069, 887, 6397, 1129, 3137, 5351, 2971, 1327, 14107, 9973, 19333, 84871, 16141, 15683, 73189, 31907, 28229, 35617, 35677, 44293, 43331, 107377, 34061, 221327, 134513, 31397, 480209, 173359, 332317, 933073, 736279, 265621, 843911, 404851, 155921

Offset

0,1

Comments

Inspired by A295973.

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

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.

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

As an example, sfd = -2 for p = 7, 13, 31, 37, 61, 67, 73, 97, 103, 157, 193, 223, 271, 277, 307, ..., .

Links

Table of n, a(n) for n=0..41.

Eric Weisstein's World of Mathematics, Forward Difference.

Example

a(0) = 3 since the three consecutive primes {3, 5, 7} have an sfd of 0;
a(1) = 7 since the three consecutive primes {7, 11, 13} have a sfd of -2;
a(2) = 23 since the three consecutive primes {23, 29, 31} have a sfd of -4;
a(3) = 1531 since the three consecutive primes {1531, 1543, 1549} have an sfd of -6;
a(4) =  since the three consecutive primes {} have an sfd of -8; etc.

Mathematica

p = 2; q = 3; r = 5; t[_] := 0; While[p < 1000000, d = p - 2q + r; If[ t[d] == 0, t[d] = p]; p = q; q = r; r = NextPrime@ r]; Array[ t[-2#] &, 42, 0]

Prog

(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

Crossrefs

Cf. A000040, A000230, A036263, A137501, A295746, A295973, A316791.

Sequence in context: A067604 A090118 A099183 * A358974 A110864 A046102

Adjacent sequences: A316789 A316790 A316791 * A316793 A316794 A316795

Keyword

nonn

Author

Edward Bernstein and Robert G. Wilson v, Jul 14 2018

Status

approved