A173177 - OEIS

A173177

Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

%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