A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

2, 5, 8, 14, 17, 20, 29, 32, 35, 38, 47, 50, 53, 62, 68, 74, 77, 80, 89, 95, 98, 104, 110, 113, 119, 134, 137, 140, 152, 155, 164, 167, 173, 182, 185, 188, 197, 203, 209, 215, 218, 227, 230, 242, 248, 260, 269, 272, 284, 287, 299

Offset: 1

Author

Eric Desbiaux, Feb 11 2010

Keywords

• nonn
• uned

Comments

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

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

first type

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

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

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

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

OR

second type

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

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

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

k a b OR a b

-- - - - -

0 0 0 0 0

1 - - - -

2 1 0 1 0

3 0 1 0 1

4 2 0 2 0

5 1 1 1 1

6 0 2 3 0

7 2 1 2 1

8 1 2 4 0

9 0 3 3 1

10 2 2 5 0

11 1 3 4 1

12 0 4 6 0

13 2 3 5 1

14 1 4 7 0

15 0 5 6 1

...

2* 2 + 3 OR 3* 1 + 4 = 7;

2* 5 + 3 OR 3* 3 + 4 = 13;

2* 8 + 3 OR 3* 5 + 4 = 19;

2*14 + 3 OR 3* 9 + 4 = 31;

2*17 + 3 OR 3*11 + 4 = 37;

2*20 + 3 OR 3*13 + 4 = 43;

2*29 + 3 OR 3*19 + 4 = 61;

2*32 + 3 OR 3*21 + 4 = 67;

2*35 + 3 OR 3*23 + 4 = 73.

A034936 Numbers k such that 3k+4 is prime.

A002476 Primes of the form 6k+1.

A024899 Nonnegative integers k such that 6k+1 is prime.

2, 5, 8, 14, 17, 20, ... = (3*(4*A024899 - A034936) - 5)/2.

Links

• Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?

Crossrefs

Cf. A067076, A034936, A002476, A024899.

Programs

Mathematica

Select[Range[300],PrimeQ[2#+3]&&Divisible[2#-1,3]&] (* Harvey P. Dale, Aug 25 2016 *)

Extensions

More terms from Harvey P. Dale, Aug 25 2016