A024897 -id:A024897 - cp's OEIS frontend

A023218 Primes p such that 5*p + 4 is also prime.

3, 5, 11, 17, 29, 47, 53, 71, 83, 89, 101, 113, 131, 167, 251, 257, 263, 281, 311, 389, 419, 461, 467, 479, 491, 509, 521, 557, 563, 587, 593, 599, 617, 641, 659, 677, 743, 797, 809, 827, 857, 881, 929, 977, 983, 1019, 1061, 1103, 1187, 1217, 1259, 1277, 1289, 1319

Offset: 1

David W. Wilson

Except for the first term, all terms are congruent to 5 (mod 6). - John Cerkan, Sep 07 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Subsequence of primes of A024897.

[n: n in [0..1000] | IsPrime(n) and IsPrime(5*n+4)]; // Vincenzo Librandi, Nov 20 2010

A023218:=n->`if`(isprime(n) and isprime(5*n+4), n, NULL): seq(A023218(n), n=1..2*10^3); # Wesley Ivan Hurt, Sep 07 2016

lst={};Do[If[PrimeQ[n]&&PrimeQ[5*n+4], AppendTo[lst, n]], {n, 13^3}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)
Select[Prime[Range[300]],PrimeQ[5#+4]&] (* Harvey P. Dale, Dec 31 2013 *)

A153343 Numbers k such that 5*k + 4 is not prime.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 76, 78, 79, 80, 82, 84, 85, 86, 88

Offset: 1

Vincenzo Librandi, Dec 24 2008

Apart from a(0) = 0 and a(1) = 1 this sequence comprises those numbers k such that (5*k)!/(5*k + 4) is an integer. - Peter Bala, Jan 25 2017

			Distribution of the odd terms in the following triangular array:
   1;
   *,   *;
   *,   *,   9;
   *,   *,   *,   *;
   *,   *,   *,  19,   *;
   7,   *,   *,   *,   *,  33;
   *,   *,   *,   *,   *,   *,   *;
   *,   *,  23,   *,   *,   *,   *,  57;
   *,   *,   *,   *,  41,   *,   *,   *,   *;
   *,   *,   *,  37,   *,   *,   *,   *,  79,   *;
  13,   *,   *,   *,   *,  59,   *,   *,   *,   *,  105;
etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h - 3)/5 with h >= k >= 1. - _Vincenzo Librandi_, Jan 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A024897, A014076, A153170, A045751, A095277, A153088, A153343.

[n: n in [0..150] | not IsPrime(5*n + 4)]; // Vincenzo Librandi, Jan 12 2013

# produces the sequence apart from the initial terms 0 and 1
for n from 0 to 100 do
if irem(factorial(5*n), 5*n+4) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017

Select[Range[0, 200], !PrimeQ[5*# + 4]&] (* Vincenzo Librandi, Jan 12 2013 *)

Erroneous comment deleted by N. J. A. Sloane, Jun 23 2010

0 added by Arkadiusz Wesolowski, Aug 03 2011

A111225 Numbers n such that 5*n + 8 is prime.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 31, 33, 37, 43, 45, 51, 55, 57, 61, 69, 73, 75, 85, 87, 91, 99, 103, 111, 117, 121, 127, 129, 133, 135, 145, 147, 153, 163, 169, 171, 175, 189, 195, 201, 205, 211, 217, 219, 223, 229, 231, 237, 241, 243, 255, 259, 273, 283, 285, 289

Offset: 1

Parthasarathy Nambi, Oct 26 2005

nonn

			If n=103 then 5*n + 8 = 523 (prime).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A024894, A087505, A024897, A081759.

[n: n in [0..100000] |IsPrime(5*n+8)]; // Vincenzo Librandi, Sep 03 2010

Select[Range[300],PrimeQ[5#+8]&] (* Harvey P. Dale, Oct 31 2011 *)

is(n)=isprime(5*n+8) \\ Charles R Greathouse IV, Feb 17 2017

A111224 Numbers n such that 5*n + 7 is prime.

Original entry on oeis.org

0, 2, 6, 8, 12, 18, 20, 24, 26, 30, 32, 38, 44, 50, 54, 60, 62, 66, 68, 72, 78, 90, 92, 96, 108, 110, 114, 116, 120, 122, 128, 134, 144, 150, 156, 158, 164, 170, 174, 176, 180, 186, 188, 192, 194, 198, 216, 218, 222, 236, 242, 246, 254, 258, 260, 264, 272, 284

Offset: 1

Parthasarathy Nambi, Oct 26 2005

			If n=108 then 5*n + 7 = 547 (prime).

Cf. A024894, A087505, A024897, A081759.

[n: n in [0..300] | IsPrime(5*n+7)]; // Vincenzo Librandi, Sep 03 2010

Select[Range[0,300],PrimeQ[5#+7]&] (* Harvey P. Dale, Jun 24 2018 *)

is(n)=isprime(5*n+7) \\ Charles R Greathouse IV, Jun 12 2017

A153355 Numbers k such that 5k-1 is a prime.

Original entry on oeis.org

4, 6, 12, 16, 18, 22, 28, 30, 36, 40, 46, 48, 54, 70, 72, 76, 78, 82, 84, 88, 90, 96, 100, 102, 114, 120, 124, 132, 142, 144, 148, 154, 162, 166, 168, 172, 184, 186, 202, 204, 208, 210, 214, 222, 226, 246, 250, 252, 256, 258, 264, 280, 282, 286, 288, 292, 298

Offset: 1

Vincenzo Librandi, Dec 24 2008

One more than the value of A024897 at the same index. - R. J. Mathar, Jan 05 2009

			5*4 - 1 = 19 is a prime, so 4 is a term;
5*30 - 1 = 149 is a prime, so 30 is a term.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

[n: n in [0..300] | IsPrime(5*n - 1)]; // Vincenzo Librandi, Sep 24 2012

Select[Range[400],PrimeQ[5#-1]&] (* Vladimir Joseph Stephan Orlovsky, Feb 25 2011*)

The set of numbers (1+A030433(k))/5, k=1,2,3,4,.... - R. J. Mathar, Jan 03 2009

Extended by R. J. Mathar, Jan 05 2009

A023314 Primes that remain prime through 4 iterations of function f(x) = 5x + 4.

Original entry on oeis.org

263, 1217, 2141, 4673, 5333, 6983, 10973, 11423, 26669, 27143, 28697, 74843, 85061, 101063, 102647, 114113, 121001, 133349, 141623, 150497, 154823, 199877, 200183, 202409, 203039, 207953, 259697, 259781, 275813, 280487, 294167, 305477, 322727

Offset: 1

David W. Wilson

nonn

Primes p such that 5*p+4, 25*p+24, 125*p+124 and 625*p+624 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023218, A023253, A023284, and A024897.

[n: n in [1..1000000] | IsPrime(n) and IsPrime(5*n+4) and IsPrime(25*n+24) and IsPrime(125*n+124) and IsPrime(625*n+624)]; // Vincenzo Librandi, Aug 04 2010

a(n) == 11 or 41 (mod 42). - John Cerkan, Oct 07 2016

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

J. M. Bergot, Mar 19 2007

nonn

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

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 *)

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

A023342 Primes that remain prime through 5 iterations of function f(x) = 5x + 4.

Original entry on oeis.org

5333, 26669, 294167, 324869, 344189, 578297, 676829, 807407, 894893, 1078559, 1114427, 1174487, 1624349, 1883363, 2247923, 2926769, 3075029, 3196871, 3427871, 3558407, 4037039, 4205879, 5392799, 6130823, 6479423, 6714497, 6750113, 6915299

Offset: 1

David W. Wilson

nonn

Primes p such that 5*p+4, 25*p+24, 125*p+124, 625*p+624 and 3125*p+3124 are also primes. - Vincenzo Librandi, Aug 05 2010

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

Subsequence of A023218, A023253, A023284, A023314, and A024897.

[n: n in [1..10000000] | IsPrime(n) and IsPrime(5*n+4) and IsPrime(25*n+24) and IsPrime(125*n+124) and IsPrime(625*n+624) and IsPrime(3125*n+3124)] // Vincenzo Librandi, Aug 05 2010

Select[Prime[Range[500000]],AllTrue[Rest[NestList[5#+4&,#,5]],PrimeQ]&] (* Harvey P. Dale, Jan 01 2022 *)

a(n) == 41 (mod 42). - John Cerkan, Oct 20 2016

A178082 Numbers k for which 5k-4, 5k-2, 5k+2, and 5k+4 are primes.

Original entry on oeis.org

3, 21, 39, 165, 297, 375, 417, 651, 693, 1131, 1887, 2601, 3129, 3147, 3213, 3609, 3783, 3885, 4203, 4455, 5061, 6345, 6969, 8757, 10269, 11067, 12597, 13443, 13899, 14445, 15453, 15939, 16209, 16545, 17763, 19569, 19827, 20223, 21969, 23307

Offset: 1

Roger L. Bagula, May 19 2010

			The associated prime quadruplets start as:
     11,    13,    17,    19;   (for n =  3)
    101,   103,   107,   109;   (for n = 21)
    191,   193,   197,   199;   (for n = 39)
    821,   823,   827,   829;
   1481,  1483,  1487,  1489;
   1871,  1873,  1877,  1879;
   2081,  2083,  2087,  2089;
   3251,  3253,  3257,  3259;
   3461,  3463,  3467,  3469;
   5651,  5653,  5657,  5659;
   9431,  9433,  9437,  9439;
  13001, 13003, 13007, 13009;
  15641, 15643, 15647, 15649;
  15731, 15733, 15737, 15739;
  16061, 16063, 16067, 16069;
  18041, 18043, 18047, 18049;
  18911, 18913, 18917, 18919;
  19421, 19423, 19427, 19429.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007811, A024897, A024896.

[n: n in [0..1000]| IsPrime(5*n - 4) and IsPrime(5*n - 2) and IsPrime(5*n + 2) and IsPrime(5*n + 4)]; // Vincenzo Librandi, Nov 30 2010

Flatten[Table[If[PrimeQ[5*n + 2] && PrimeQ[5*n - 2] && PrimeQ[5*n + 4] && PrimeQ[5*n - 4], n, {}], {n, 0, 10000}]]
Select[Range[25000],AllTrue[5#+{4,2,-2,-4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 03 2018 *)

a(n) = A173037(n+1)/5.

A111226 Numbers n such that 5*n + 12 is prime.

Original entry on oeis.org

1, 5, 7, 11, 17, 19, 23, 25, 29, 31, 37, 43, 49, 53, 59, 61, 65, 67, 71, 77, 89, 91, 95, 107, 109, 113, 115, 119, 121, 127, 133, 143, 149, 155, 157, 163, 169, 173, 175, 179, 185, 187, 191, 193, 197, 215, 217, 221, 235, 241, 245, 253, 257, 259, 263, 271, 283, 287

Offset: 1

Parthasarathy Nambi, Oct 26 2005

			If n=109 then 5*n + 12 = 557 (prime).

Cf. A024894, A087505, A024897, A081759.

[n: n in [0..100000] |IsPrime(5*n+12)]; // Vincenzo Librandi, Nov 13 2010

Mathematica
```
Select[Range[300], PrimeQ[5 # + 12] &]
```

is(n)=isprime(5*n+12) \\ Charles R Greathouse IV, Jun 13 2017

cp's OEIS Frontend

A023218 Primes p such that 5*p + 4 is also prime.

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A153343 Numbers k such that 5*k + 4 is not prime.

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A111225 Numbers n such that 5*n + 8 is prime.

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Magma

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A111224 Numbers n such that 5*n + 7 is prime.

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Magma

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PARI

A153355 Numbers k such that 5k-1 is a prime.

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Magma

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A023314 Primes that remain prime through 4 iterations of function f(x) = 5x + 4.

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Magma

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

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Mathematica

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A023342 Primes that remain prime through 5 iterations of function f(x) = 5x + 4.

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A178082 Numbers k for which 5k-4, 5k-2, 5k+2, and 5k+4 are primes.