A034936 -id:A034936 - cp's OEIS frontend

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

Eric Desbiaux, Feb 11 2010

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.

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.

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

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

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

Data corrected and extended by Amiram Eldar, Jul 30 2024

A023336 Primes that remain prime through 5 iterations of function f(x) = 3x + 4.

Original entry on oeis.org

34613, 165443, 321053, 363403, 474143, 496333, 528673, 631853, 834503, 957563, 1199623, 1274803, 1817093, 1918733, 2063423, 2611663, 2889703, 3224233, 3652703, 3697433, 3824413, 3852973, 4655873, 4708793, 5089943, 5508263, 5937853, 6067073

Offset: 1

David W. Wilson

nonn

Primes p such that 3*p+4, 9*p+16, 27*p+52, 81*p+160 and 243*p+484 are also primes. - Vincenzo Librandi, Aug 05 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023209, A023247, A023278, A023308, and A034936.

[n: n in [1..25000000] | IsPrime(n) and IsPrime(3*n+4) and IsPrime(9*n+16) and IsPrime(27*n+52) and IsPrime(81*n+160) and IsPrime(243*n+484)] // Vincenzo Librandi, Aug 05 2010

a(n) == 33 (mod 70). - John Cerkan, Oct 10 2016

A124855 Numbers k such that 3k + 4 and 4k + 3 are primes.

Original entry on oeis.org

1, 5, 11, 19, 25, 31, 41, 49, 59, 65, 89, 91, 109, 115, 121, 125, 151, 161, 179, 181, 205, 209, 229, 241, 245, 275, 305, 329, 331, 349, 355, 361, 371, 389, 415, 439, 509, 515, 521, 535, 551, 595, 599, 625, 661, 665, 671, 719, 725, 749, 755, 769, 779, 791, 839

Offset: 1

Zak Seidov, Nov 10 2006

nonn

Intersection of A034936 and A095278. Prime terms are in A106068.

[n: n in [0..1000] | IsPrime(3*n+4) and IsPrime(4*n+3)] // Vincenzo Librandi, Mar 26 2010

Select[Range[850], PrimeQ[3# + 4] && PrimeQ[4# + 3] &] (* Ray Chandler, May 11 2007 *)

A154618 Triangle read by rows: integer values of T(n,m) = (4mn+2m+2n-3)/3.

Original entry on oeis.org

7, 15, 17, 39, 29, 55, 27, 61, 95, 43, 81, 119, 37, 83, 129, 175, 57, 107, 157, 207, 47, 105, 163, 221, 279, 71, 133, 195, 257, 319, 57, 127, 197, 267, 337, 407, 85, 159, 233, 307, 381, 455, 67, 149, 231, 313, 395, 477, 559, 99, 185, 271, 357, 443, 529, 615, 77

Offset: 1

Vincenzo Librandi, Jan 16 2009

			The sequence contains the integers selected from the full table:
5/3;
11/3,7;
17/3,31/3,15;
23/3,41/3,59/3,77/3;
29/3,17,73/3,95/3,39;
35/3,61/3,29,113/3,139/3,55;
41/3,71/3,101/3,131/3,161/3,191/3,221/3;
47/3,27,115/3,149/3,61,217/3,251/3,95;

Cf. A034936, A154616

Select[Flatten[Table[(4*m*n+2*m+2*n-3)/3,{m,30},{n,m}]],IntegerQ] (* Harvey P. Dale, Jun 10 2015 *)

Keyword:tabl removed, appearance of fractions clarified by R. J. Mathar, Oct 16 2009

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A023336 Primes that remain prime through 5 iterations of function f(x) = 3x + 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

A124855 Numbers k such that 3k + 4 and 4k + 3 are primes.

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

A154618 Triangle read by rows: integer values of T(n,m) = (4mn+2m+2n-3)/3.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A023336 Primes that remain prime through 5 iterations of function f(x) = 3x + 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

A124855 Numbers k such that 3k + 4 and 4k + 3 are primes.

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

A154618 Triangle read by rows: integer values of T(n,m) = (4*m*n+2*m+2*n-3)/3.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

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

A154618 Triangle read by rows: integer values of T(n,m) = (4mn+2m+2n-3)/3.