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%I #14 Apr 03 2023 10:36:11
%S 2,5,8,14,17,20,29,32,35,38,47,50,53,62,68,74,77,80,89,95,98,104,110,
%T 113,119,134,137,140,152,155,164,167,173,182,185,188,197,203,209,215,
%U 218,227,230,242,248,260,269,272,284,287,299
%N Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.
%C With 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
%C 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* 2 + 3 OR 3* 1 + 4 = 7;
%C 2* 5 + 3 OR 3* 3 + 4 = 13;
%C 2* 8 + 3 OR 3* 5 + 4 = 19;
%C 2*14 + 3 OR 3* 9 + 4 = 31;
%C 2*17 + 3 OR 3*11 + 4 = 37;
%C 2*20 + 3 OR 3*13 + 4 = 43;
%C 2*29 + 3 OR 3*19 + 4 = 61;
%C 2*32 + 3 OR 3*21 + 4 = 67;
%C 2*35 + 3 OR 3*23 + 4 = 73.
%C A034936 Numbers k such that 3k+4 is prime.
%C A002476 Primes of the form 6k+1.
%C A024899 Nonnegative integers k such that 6k+1 is prime.
%C 2, 5, 8, 14, 17, 20, ... = (3*(4*A024899 - A034936) - 5)/2.
%H Prime FAQ Chris K.Caldwell, <a href="https://t5k.org/notes/faq/six.html">Most rediscovered result about primes numbers</a>
%t Select[Range[300],PrimeQ[2#+3]&&Divisible[2#-1,3]&] (* _Harvey P. Dale_, Aug 25 2016 *)
%Y Cf. A067076, A034936, A002476, A024899.
%K nonn,uned
%O 1,1
%A _Eric Desbiaux_, Feb 11 2010
%E More terms from _Harvey P. Dale_, Aug 25 2016
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