%I #22 Jul 30 2024 14:35:54

%S 1,4,7,10,13,19,22,25,28,34,40,43,49,52,55,64,67,73,82,85,88,94,97,

%T 112,115,118,124,127,130,133,139,145,154,157,172,175,178,190,193,199,

%U 208,214,220,223,229,232,238,244,250,253,259,277,280,283,292,295,298,307,319

%N Numbers k such that 2*k+3 is a prime of the form 3*A024893(m) + 2.

%C With the Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,

%C for k > 1, k = 2*a + 3*b (a and b integers)

%C first type

%C A001477 = (2*A080425) + (3*A008611)

%C A000040 = (2*A039701) + (3*A157966)

%C A024893 Numbers k such that 3*k + 2 is prime

%C A034936 Numbers k such that 3*k + 4 is prime

%C OR second type

%C A001477 = (2*A028242) + (3*A059841)

%C A000040 = (2*A067076) + (3*1)

%C A067076 Numbers k such that 2*k + 3 is prime

%C k a b OR a b

%C -- - - - -

%C 0 0 0 0 0

%C 1 - - - -

%C 2 1 0 1 0

%C 3 0 1 0 1

%C 4 2 0 2 0

%C 5 1 1 1 1

%C 6 0 2 3 0

%C 7 2 1 2 1

%C 8 1 2 4 0

%C 9 0 3 3 1

%C 10 2 2 5 0

%C 11 1 3 4 1

%C 12 0 4 6 0

%C 13 2 3 5 1

%C 14 1 4 7 0

%C 15 0 5 6 1

%C ...

%C 2* 1 + 3 OR 3* 1 + 2 = 5;

%C 2* 4 + 3 OR 3* 3 + 2 = 11;

%C 2* 7 + 3 OR 3* 5 + 2 = 17;

%C 2*10 + 3 OR 3* 7 + 2 = 23;

%C 2*13 + 3 OR 3* 9 + 2 = 29;

%C 2*19 + 3 OR 3*13 + 2 = 41;

%C 2*22 + 3 OR 3*15 + 2 = 47;

%C 2*25 + 3 OR 3*17 + 2 = 53;

%C 2*28 + 3 OR 3*19 + 2 = 59.

%C A024893 Numbers k such that 3k+2 is prime.

%C A007528 Primes of the form 6k-1.

%C A024898 Positive integers k such that 6k-1 is prime.

%C 1, 4, 7, 10, 13, 19, ... = (3*(4*A024898 - A024893) - 7)/2 = (A112774 - 3*A024893 - 5)/2 = A003627 - (3*A024893 - 5)/2.

%H Amiram Eldar, <a href="/A173178/b173178.txt">Table of n, a(n) for n = 1..10000</a>

%H Chris K. Caldwell, <a href="https://t5k.org/notes/faq/six.html">FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?</a>, Frequently asked questions about primes.

%F a(n) = 3*A059325(n) + 1. - _Amiram Eldar_, Jul 30 2024

%t Select[Range[0, 320], PrimeQ[(p = 2*# + 3)] && Mod[p, 3] == 2 &] (* _Amiram Eldar_, Jul 30 2024 *)

%Y Cf. A067076, A024893, A007528, A024898, A059325.

%K nonn,easy,uned

%O 1,2

%A _Eric Desbiaux_, Feb 11 2010

%E Data corrected and extended by _Amiram Eldar_, Jul 30 2024