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

1,1

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.

Links

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

Formula

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A002145, A002144, A158709, A349667.

Sequence in context: A000921 A185004 A172490 * A298039 A376828 A135659

Adjacent sequences: A349663 A349664 A349665 * A349667 A349668 A349669

Keyword

nonn

Author

Karl-Heinz Hofmann, Dec 23 2021

Status

approved