cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-15 of 15 results.

A126956 Numbers n such that 3n+2, 4n+3 and 5n+4 are primes.

Original entry on oeis.org

5, 17, 77, 89, 119, 185, 257, 287, 395, 665, 755, 797, 929, 1175, 1259, 1337, 1379, 1445, 1469, 1769, 2057, 2105, 3125, 3419, 3437, 3629, 3815, 3989, 4079, 4157, 4175, 4217, 4367, 4445, 4847, 5045, 5375, 6089, 6137, 6167, 6359, 6419, 6485, 6725, 6887
Offset: 1

Views

Author

J. M. Bergot, Mar 19 2007

Keywords

Examples

			Take n = 185. Then 3*185 + 2 = 557, 4*185 + 3 = 743 and 5*185 + 4 = 929 are primes.

Links

Crossrefs

Intersection of A024893, A095278, A024897. Cf. A126955.

Programs

  • Mathematica
    Select[Range[7000], PrimeQ[3# + 2] && PrimeQ[4# + 3] && PrimeQ[5# + 4] &] (* Ray Chandler, Mar 20 2007 *)
Select[Range[7000],AllTrue[{3#+2,4#+3,5#+4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 06 2019 *)

Extensions

Corrected and extended by Ray Chandler, Stuart Clary, Robert G. Wilson v and Zak Seidov, Mar 20 2007

A023307 Primes that remain prime through 4 iterations of function f(x) = 3x + 2.

Original entry on oeis.org

1129, 10009, 11489, 12539, 13859, 30029, 63079, 77359, 99119, 121039, 124669, 169409, 194749, 205589, 246329, 330329, 349519, 351829, 354839, 361279, 369539, 384589, 395719, 399769, 416989, 429109, 446819, 527599, 532489, 544259, 575119
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Primes p such that 3*p+2, 9*p+8, 27*p+26, and 81*p+80 are also primes. - Vincenzo Librandi, Aug 04 2010

Links

Crossrefs

Subsequence of A023208, A023246, A023277, and A024893.

Programs

  • Magma
    [n: n in [1..1000000] | IsPrime(n) and IsPrime(3*n+2) and IsPrime(9*n+8) and IsPrime(27*n+26) and IsPrime(81*n+80)] // Vincenzo Librandi, Aug 04 2010
  • Mathematica
    Select[Prime[Range[50000]],AllTrue[Rest[NestList[3#+2&,#,4]],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 13 2015 *)

Formula

a(n) == 9 or 69 (mod 70). - John Cerkan, Oct 04 2016

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

Original entry on oeis.org

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

Views

Author

Eric Desbiaux, Feb 11 2010

Keywords

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

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

A268475 Numbers n such that n^3 +/- 2 and 3*n +/- 2 are all prime.

Original entry on oeis.org

435, 555, 2415, 31635, 38025, 44835, 80625, 88335, 97455, 98505, 99435, 124335, 142065, 145095, 165375, 176055, 204765, 246435, 279225, 293475, 310095, 315555, 332085, 344745, 348735, 376935, 392415, 443595, 462105, 467385, 482355, 581415, 609555, 626775, 636015
Offset: 1

Views

Author

K. D. Bajpai, Feb 05 2016

Keywords

Comments

All the terms in this sequence are congruent to 0 (mod 5).
Each term in this sequence yields two sets of cousin prime pairs i.e., for n = 435 -> {82312877, 82312873} and {1307, 1303}.
All terms are congruent to 15 mod 30. - Robert Israel, Feb 05 2016

Examples

			435 is in the sequence because 435^3 + - 2 =  82312877, 82312873; 3*435 + - 2 = 1307, 1303 are all prime.
555 is in the sequence because 555^3 + - 2 =  170953877, 170953873; 3*555 + - 2 = 1667, 1663 are all prime.

Links

Crossrefs

Cf. A024893, A090121, A108701, A153183, A157834.

Programs

  • Magma
    [n : n in [1..1e5] | IsPrime(n^3 + 2) and IsPrime(n^3 - 2) and IsPrime(3*n + 2) and IsPrime(3*n - 2)];
  • Maple
    select(n -> andmap(isprime, [n^3 + 2, n^3 - 2, 3*n + 2, 3*n - 2]), [seq(p, p=1.. 10^6)]);
  • Mathematica
    Select[Range[1000000], PrimeQ[#^3 + 2] && PrimeQ[#^3 - 2] && PrimeQ[3 # + 2] && PrimeQ[3 # - 2] &]
  • PARI
    for(n = 1,1e5, if( isprime(n^3 + 2) && isprime(n^3 - 2) && isprime(3*n + 2) && isprime(3*n - 2), print1(n ", ")))

A294064 Numbers k such that 2*k - 3, 2*k + 3, 3*k - 2, 3*k + 2 are primes.

Original entry on oeis.org

5, 7, 13, 35, 43, 55, 77, 127, 133, 155, 167, 253, 287, 295, 365, 475, 497, 533, 595, 713, 1007, 1177, 1483, 1805, 2323, 2575, 2723, 2927, 3107, 3415, 3487, 3823, 4145, 4213, 4367, 4565, 4717, 4927, 4963, 5125, 5215, 5363, 5417, 5587, 5627, 5795, 6133, 6587, 6797
Offset: 1

Views

Author

Dimitris Valianatos, Oct 22 2017

Keywords

Comments

The common numbers of A098090, A067076, A153183, A024893.
Conjecture: The Sum_{n>=1} 1/a(n) = 0.57... converges.
Note that the sum of the 4 primes that are obtained is 10 times the original term: (2*k - 3) + (2*k + 3) + (3*k - 2) + (3*k + 2) = 10*k.
From Robert G. Wilson v, Nov 19 2017: (Start)
Number of terms less than 10^m: 2, 7, 20, 55, 189, 919, 4863, 28218, 174469, ..., ;
Number of prime terms less than 10^m: 2, 4, 6, 12, 39, 140, 558, 2755, 14804, ..., .
All terms are == {5, 7, 13, 17, 23, 25} (mod 30).
(End)

Examples

			5 is in the sequence because 2*5-3 = 7, 2*5+3 = 13, 3*5-2 = 13, 3*5+2 = 17 and the tetrad [7, 13, 13, 17] are all prime numbers.
7 is in the sequence because 2*7-3 = 11, 2*7+3 = 17, 3*7-2 = 19, 3*7+2 = 23 and the tetrad [11, 17, 19, 23] are all prime numbers.

Links

Crossrefs

Cf. A098090, A067076, A153183, A024893.

Programs

  • Mathematica
    Select[Range[10^4], Function[k, AllTrue[Flatten@ Map[#1 k + {-1, 1} #2 & @@ # &, {#, Reverse@ #}] &@ {2, 3}, PrimeQ]]] (* Michael De Vlieger, Oct 22 2017 *)
  • PARI
    {
for(n=1,10000,
    if(isprime(2*n-3)&&isprime(2*n+3)&&isprime(3*n-2)&&isprime(3*n+2),
       print1(n", ")
      )
   )
}
Previous Showing 11-15 of 15 results.

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