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

%I #33 Mar 17 2022 17:20:30

%S 7,31,47,71,127,151,167,239,311,431,439,479,607,631,647,727,839,911,

%T 967,991,1039,1231,1319,1399,1471,1511,1559,1567,1607,1879,1951,1999,

%U 2111,2239,2311,2351,2447,2671,2719,2927,3119,3167,3191,3359,3391,3671,3727,3767,3911

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

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

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

%C 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.

%C Apparently this is a subsequence of A158709.

%H Karl-Heinz Hofmann, <a href="/A349666/b349666.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) == 7 (mod 8). - _Hugo Pfoertner_, Dec 25 2021

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

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

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

%Y Cf. A002145, A002144, A158709, A349667.

%K nonn

%O 1,1

%A _Karl-Heinz Hofmann_, Dec 23 2021