A235914 Odd primes p = 2*m + 1 with m*(m-1) - prime(m) and m*(m+1) - prime(m) both prime.

13, 17, 23, 29, 31, 43, 73, 89, 181, 229, 313, 367, 379, 557, 631, 683, 1021, 1069, 1093, 1151, 1303, 1459, 1471, 1663, 1733, 1831, 1871, 2411, 2473, 2791, 2843, 2887, 3673, 3691, 3793, 3797, 3863, 4001, 4139, 4261, 5261, 5431, 6091, 6301, 6661, 6737, 6883, 7489, 7523, 7873

Offset: 1

Author

Zhi-Wei Sun, Jan 16 2014

Keywords

• nonn

Comments

By the conjecture in A235912, this sequence should have infinitely many terms.

Examples

			a(1) = 13 since none of 1*2 - prime(1) = 0, 1*2 - prime(2) = -1, 2*3 - prime(3) = 1 and 2*4 + 1 = 9 = 4*5 - prime(5) is prime, but 2*6 + 1 = 13, 5*6 - prime(6) = 30 - 13 = 17 and 6*7 - prime(6) = 42 - 13 = 29 are all prime.
		

Links

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

Crossrefs

Cf. A000040, A235727, A235912.

Programs

Mathematica

PQ[n_]:=n>0&&PrimeQ[n]
q[n_]:=PQ[n(n-1)-Prime[n]]&&PQ[n(n+1)-Prime[n]]
n=0;Do[If[q[(Prime[k]-1)/2],n=n+1;Print[n," ",Prime[k]]],{k,2,1000}]