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A136079 Father primes of order 10.

%I A136079 #25 Jan 30 2025 05:35:30
%S A136079 83,167,251,293,419,503,797,881,1259,1301,1427,1511,1553,1889,2141,
%T A136079 2267,2309,2393,2687,2897,2939,3191,3527,3779,3821,4073,4157,4451,
%U A136079 4703,4787,5039,5081,5417,5669,5711,6173,6551,6971,7307,7349,7433,7559,7727,7853
%N A136079 Father primes of order 10.
%C A136079 For smallest father primes of order n, see A136026 (also definition). For father primes of orders 1,2,...,9, see A094524, A136071, A136072, A136073, A136074, A136075, A136076, A136077, A136078, respectively.
%C A136079 From _Bob Selcoe_, Apr 25 2014: (Start)
%C A136079 In general, a father prime, p', of order k is of the form p'=2k+(2k+1)*p for some prime, p. In this sequence, k=10, and so each prime is of the form p'=20+21p where p ranges over {3,7,11,13,19,23,...}. Thus a father prime p' has order k when (p'-2k)/(2k+1) is prime.
%C A136079 Father primes (p') of order k will be of the form: p'(mod (4k+2))=4k+1, or p'=(4k+2)*j-1, j>=2. For this sequence: k=10, 4k+2=42; j={2,4,6,7,10,12,...}. So for example, j=7 generates a father prime because 42*7-1 = 293 AND (293-(2*10))/(2*10+1) = 13, since both 13 and 293 are prime. Note that not all j such that (4k+2)*j-1 is prime will produce a father prime. In this example, when j=11, 42*11-1=461 (prime); but (461-(2*10))/(2*10+1) = 21 (not prime). (End)
%H A136079 Amiram Eldar, <a href="/A136079/b136079.txt">Table of n, a(n) for n = 1..10000</a>
%t A136079 n = 10; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a
%Y A136079 Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136076, A136077, A136078, A136080.
%K A136079 nonn
%O A136079 1,1
%A A136079 _Artur Jasinski_, Dec 12 2007