A232109 Least prime p < n + 5 with n + (p-1)*(p-3)/8 prime, or 0 if such a prime p does not exist.

5, 3, 3, 5, 3, 5, 3, 7, 11, 5, 3, 5, 3, 7, 17, 5, 3, 5, 3, 7, 11, 5, 3, 23, 17, 7, 11, 5, 3, 5, 3, 13, 11, 7, 19, 5, 3, 7, 17, 5, 3, 5, 3, 7, 17, 5, 3, 23, 11, 7, 11, 5, 3, 23, 17, 7, 11, 5, 3, 5, 3, 31, 11, 7, 19, 5, 3, 7, 11, 5, 3, 5, 3, 13, 17, 7, 19, 5, 3, 7, 17, 5, 3, 23, 17, 7, 11, 5, 3, 29, 11, 13, 11, 7, 19, 5, 3, 7, 11, 5

Offset

1,1

Comments

Conjecture: a(n) > 0 for all n > 0. Moreover, for any integer n > 1 there exists a prime p < 2*sqrt(n)*log(7n) such that n + (p-1)*(p-3)/8 is prime.

This implies that any integer n > 1 can be written as (p-1)/2 + q with q a positive integer, and p and (p^2-1)/8 + q both prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(1) = 5 since neither 1 + (2-1)*(2-3)/8 = 7/8 nor 1 + (3-1)*(3-3)/8 = 1  is prime, but 1 + (5-1)*(5-3)/8 = 2 is prime.

Mathematica

Do[Do[If[PrimeQ[n+(Prime[k]-1)(Prime[k]-3)/8], Goto[aa]], {k, 1, PrimePi[n+4]}];
Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 100}]

Crossrefs

Cf. A000040, A185636, A204065, A227898, A228425, A231201, A231557.

Sequence in context: A200099 A182129 A010038 * A270753 A198923 A056597

Adjacent sequences: A232106 A232107 A232108 * A232110 A232111 A232112

Keyword

nonn

Author

Zhi-Wei Sun, Nov 18 2013

Status

approved