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

1, 4, 7, 10, 13, 19, 22, 25, 28, 34, 40, 43, 49, 52, 55, 64, 67, 73, 82, 85, 88, 94, 97, 112, 115, 118, 124, 127, 130, 133, 139, 145, 154, 157, 172, 175, 178, 190, 193, 199, 208, 214, 220, 223, 229, 232, 238, 244, 250, 253, 259, 277, 280, 283, 292, 295, 298, 307, 319

Offset: 1

Author

Eric Desbiaux, Feb 11 2010

Keywords

• nonn
• easy
• uned

Comments

With the 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* 1 + 3 OR 3* 1 + 2 = 5;

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

2* 7 + 3 OR 3* 5 + 2 = 17;

2*10 + 3 OR 3* 7 + 2 = 23;

2*13 + 3 OR 3* 9 + 2 = 29;

2*19 + 3 OR 3*13 + 2 = 41;

2*22 + 3 OR 3*15 + 2 = 47;

2*25 + 3 OR 3*17 + 2 = 53;

2*28 + 3 OR 3*19 + 2 = 59.

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

A007528 Primes of the form 6k-1.

A024898 Positive integers k such that 6k-1 is prime.

1, 4, 7, 10, 13, 19, ... = (3*(4*A024898 - A024893) - 7)/2 = (A112774 - 3*A024893 - 5)/2 = A003627 - (3*A024893 - 5)/2.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?, Frequently asked questions about primes.

Crossrefs

Cf. A067076, A024893, A007528, A024898, A059325.

Programs

Mathematica

Select[Range[0, 320], PrimeQ[(p = 2*# + 3)] && Mod[p, 3] == 2 &] (* Amiram Eldar, Jul 30 2024 *)

Formula

a(n) = 3*A059325(n) + 1. - Amiram Eldar, Jul 30 2024

Extensions

Data corrected and extended by Amiram Eldar, Jul 30 2024