cp's OEIS Frontend

A173178 Numbers k such that 2*k+3 is a prime of the form 3*A024893(m) + 2.

%I A173178 #22 Jul 30 2024 14:35:54
%S A173178 1,4,7,10,13,19,22,25,28,34,40,43,49,52,55,64,67,73,82,85,88,94,97,
%T A173178 112,115,118,124,127,130,133,139,145,154,157,172,175,178,190,193,199,
%U A173178 208,214,220,223,229,232,238,244,250,253,259,277,280,283,292,295,298,307,319
%N A173178 Numbers k such that 2*k+3 is a prime of the form 3*A024893(m) + 2.
%C A173178 With the Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
%C A173178 for k > 1, k = 2*a + 3*b (a and b integers)
%C A173178 first type
%C A173178 A001477 = (2*A080425) + (3*A008611)
%C A173178 A000040 = (2*A039701) + (3*A157966)
%C A173178 A024893 Numbers k such that 3*k + 2 is prime
%C A173178 A034936 Numbers k such that 3*k + 4 is prime
%C A173178 OR second type
%C A173178 A001477 = (2*A028242) + (3*A059841)
%C A173178 A000040 = (2*A067076) + (3*1)
%C A173178 A067076 Numbers k such that 2*k + 3 is prime
%C A173178    k   a b OR a b
%C A173178   --   - -    - -
%C A173178    0   0 0    0 0
%C A173178    1   - -    - -
%C A173178    2   1 0    1 0
%C A173178    3   0 1    0 1
%C A173178    4   2 0    2 0
%C A173178    5   1 1    1 1
%C A173178    6   0 2    3 0
%C A173178    7   2 1    2 1
%C A173178    8   1 2    4 0
%C A173178    9   0 3    3 1
%C A173178   10   2 2    5 0
%C A173178   11   1 3    4 1
%C A173178   12   0 4    6 0
%C A173178   13   2 3    5 1
%C A173178   14   1 4    7 0
%C A173178   15   0 5    6 1
%C A173178   ...
%C A173178   2* 1 + 3 OR 3* 1 + 2 =  5;
%C A173178   2* 4 + 3 OR 3* 3 + 2 = 11;
%C A173178   2* 7 + 3 OR 3* 5 + 2 = 17;
%C A173178   2*10 + 3 OR 3* 7 + 2 = 23;
%C A173178   2*13 + 3 OR 3* 9 + 2 = 29;
%C A173178   2*19 + 3 OR 3*13 + 2 = 41;
%C A173178   2*22 + 3 OR 3*15 + 2 = 47;
%C A173178   2*25 + 3 OR 3*17 + 2 = 53;
%C A173178   2*28 + 3 OR 3*19 + 2 = 59.
%C A173178 A024893 Numbers k such that 3k+2 is prime.
%C A173178 A007528 Primes of the form 6k-1.
%C A173178 A024898 Positive integers k such that 6k-1 is prime.
%C A173178 1, 4, 7, 10, 13, 19, ... = (3*(4*A024898 - A024893) - 7)/2 = (A112774 - 3*A024893 - 5)/2 = A003627 - (3*A024893 - 5)/2.
%H A173178 Amiram Eldar, <a href="/A173178/b173178.txt">Table of n, a(n) for n = 1..10000</a>
%H A173178 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 A173178 a(n) = 3*A059325(n) + 1. - _Amiram Eldar_, Jul 30 2024
%t A173178 Select[Range[0, 320], PrimeQ[(p = 2*# + 3)] && Mod[p, 3] == 2 &] (* _Amiram Eldar_, Jul 30 2024 *)
%Y A173178 Cf. A067076, A024893, A007528, A024898, A059325.
%K A173178 nonn,easy,uned
%O A173178 1,2
%A A173178 _Eric Desbiaux_, Feb 11 2010
%E A173178 Data corrected and extended by _Amiram Eldar_, Jul 30 2024