A372113 Numbers k for which (k-1)/2 and 2*k+1 are both primes.

5, 11, 15, 23, 35, 39, 63, 75, 83, 95, 119, 135, 179, 215, 219, 299, 303, 315, 359, 363, 455, 459, 483, 515, 543, 615, 663, 699, 719, 735, 779, 803, 879, 915, 923, 935, 975, 999, 1019, 1043, 1143, 1155, 1175, 1199, 1295, 1323, 1355, 1383, 1439, 1539, 1595, 1659, 1679, 1755, 1763, 1815, 1859, 1883

Offset: 1

Author

Alexandre Herrera, Apr 19 2024

Keywords

• nonn

Comments

Intersection of A072055 and A104635.

Examples

			5 is a term because (5-1)/2 = 2 is prime and 2*5+1 = 11 is prime.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A072055, A104635.

Cf. A023213, A126330.

Programs

Mathematica

Select[Range[1, 2000, 2], AllTrue[{(# - 1)/2, 2 # + 1}, PrimeQ] &] (* Michael De Vlieger, Apr 19 2024 *)

Python

from sympy import isprime
def a(n): return n%2 == 1 and isprime((n-1)>>1) and isprime(2*n+1)
print([n for n in range(2, 1900) if a(n)])

Formula

a(n) = 2*A023213(n) + 1.

a(n) = (A126330(n)-1)/2.