A270384 Primes p such that (3/4)(p + 1) - 1 is also prime.

3, 7, 23, 31, 71, 79, 151, 199, 223, 239, 263, 311, 359, 463, 479, 599, 743, 751, 823, 863, 911, 991, 1031, 1063, 1103, 1151, 1303, 1399, 1471, 1583, 1759, 1823, 1831, 1879, 1999, 2111, 2143, 2311, 2383, 2503, 2543, 2551, 2663, 2671, 2719, 3023, 3079, 3119, 3191, 3391, 3511

Offset: 1

Author

Alonso del Arte, Mar 15 2016

Keywords

• nonn

Comments

Set q = (3/4)(p + 1) - 1, then (q + 1)/(p + 1) = 3/4. If this sequence is proved to be infinite, that would prove two specific cases of the Schinzel-Sierpiński conjecture regarding rational numbers.

In fact this sequence is infinite under ('merely') Dickson's conjecture, as it requires infinitely many n with 3n + 2 and 4n + 3 both prime. - Charles R Greathouse IV, Apr 01 2016

Examples

			3 is in the sequence because 3/4 * 4 - 1 = 2, which is also prime.
7 is in the sequence because 3/4 * 8 - 1 = 5, which is also prime.
11 is not in the sequence because 3/4 * 12 - 1 = 8 = 2^3.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A158721.

Programs

Mathematica

Select[Prime[Range[500]], PrimeQ[(3/4)(# + 1) - 1] &]

PARI

is(n)=n%4==3 && isprime(n\4*3+2) && isprime(n) \\ Charles R Greathouse IV, Apr 01 2016