A349666 Primes of the form 4*k+3 that are still a prime of the form 4*k+3 after 2 Collatz steps.

7, 31, 47, 71, 127, 151, 167, 239, 311, 431, 439, 479, 607, 631, 647, 727, 839, 911, 967, 991, 1039, 1231, 1319, 1399, 1471, 1511, 1559, 1567, 1607, 1879, 1951, 1999, 2111, 2239, 2311, 2351, 2447, 2671, 2719, 2927, 3119, 3167, 3191, 3359, 3391, 3671, 3727, 3767, 3911

Offset: 1

Author

Karl-Heinz Hofmann, Dec 23 2021

Keywords

• nonn

Comments

The two Collatz steps are 3*x + 1 and x/2.

Terms are primes in A002145 which after 2 Collatz iterations are still a prime in A002145.

Pythagorean primes (A002144), which are of the form 4*k+1, never produce any prime after those 2 steps. But further reducing by 2 produces primes in A349667.

Apparently this is a subsequence of A158709.

Examples

			(31*3 + 1)/2 = 47. Both 31 and 47 are primes of the form 4*k+3. Thus 31 is a term.
		

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002145, A002144, A158709, A349667.

Programs

Mathematica

Select[4*Range[0, 1000] + 3, PrimeQ[#] && Mod[(q = (3*# + 1)/2), 4] == 3 && PrimeQ[q] &] (* Amiram Eldar, Dec 24 2021 *)

PARI

isok(p) = isprime(p) && ((p%4)==3) && isprime(q=(3*p+1)/2) && ((q%4)==3); \\ Michel Marcus, Dec 23 2021

Formula

a(n) == 7 (mod 8). - Hugo Pfoertner, Dec 25 2021