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%I #13 Apr 03 2023 10:36:11
%S 0,1,4,7,10,13,19,22,25,28,34,40,43,49,52,64,67,73,82,85,88,94,97,112,
%T 115,118,124,127,130,133
%N Numbers k such that 2k+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 Prime FAQ Chris K.Caldwell, <a href="https://t5k.org/notes/faq/six.html">Most rediscovered result about primes numbers</a>
%Y Cf. A067076, A024893, A007528, A024898.
%K nonn,uned,more
%O 1,3
%A _Eric Desbiaux_, Feb 11 2010
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