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.

Showing 1-7 of 7 results.

A045751 Numbers k such that 4*k + 1 is not prime.

Original entry on oeis.org

0, 2, 5, 6, 8, 11, 12, 14, 16, 17, 19, 20, 21, 23, 26, 29, 30, 31, 32, 33, 35, 36, 38, 40, 41, 42, 44, 46, 47, 50, 51, 52, 53, 54, 55, 56, 59, 61, 62, 63, 65, 66, 68, 71, 72, 74, 75, 76, 77, 80, 81, 82, 83, 85, 86, 89, 90, 91, 92, 94, 95, 96, 98, 101, 103, 104, 106, 107, 109
Offset: 1

Views

Author

Felice Russo

Keywords

Comments

Terms (except 0) can be written as 4xy +- (x+y) for x > 0, y > 0. - Ron R Spencer, Jul 28 2016
Numbers k such that (4*k)!/(4*k + 1) is an integer. - Peter Bala, Jan 25 2017

Examples

			Distribution of the positive terms in the following triangular array:
   2;
   *,   6;
   5,   *,  12;
   *,  11,   *,  20;
   8,   *,  19,   *,  30;
   *,  16,   *,  29,   *,  42;
  11,   *,  26,   *,  41,   *,  56;
   *,  21,   *,  38,   *,  55,   *,  72;
  14,   *,  33,   *,  52,   *,  71,   *,  90;
   *,  26,   *,  47,   *,  68,   *,  89,   *, 110;
  17,   *,  40,   *,  63,   *,  86,   *, 109,   *, 132;
etc., where * marks the noninteger values of (2*h*k + k + h)/2 with h >= k >= 1. - _Vincenzo Librandi_, Jan 14 2013

Links

Crossrefs

Cf. A014076, A153170, A095277, A153088, A153329, A153343.

Programs

  • Magma
    [n: n in [0..220]| not IsPrime(4*n+1)]; // Vincenzo Librandi, Jan 28 2011
  • Maple
    for n from 0 to 100 do
if irem(factorial(4*n), 4*n+1) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017
  • Mathematica
    Select[Range[0, 200], ! PrimeQ[4 # + 1] &]
  • PARI
    is(n)=!isprime(4*n+1) \\ Charles R Greathouse IV, Jul 29 2016

Extensions

More terms from Erich Friedman

A095277 Numbers k such that 4k + 3 is composite.

Original entry on oeis.org

3, 6, 8, 9, 12, 13, 15, 18, 21, 22, 23, 24, 27, 28, 29, 30, 33, 35, 36, 38, 39, 42, 43, 45, 46, 48, 50, 51, 53, 54, 57, 58, 60, 61, 63, 64, 66, 68, 69, 71, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 85, 87, 88, 90, 92, 93, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 108
Offset: 1

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Comments

Terms can be written as (4xy +- (x-y)) - 1 for x > 0, y > 0. - Ron R Spencer, Aug 01 2016
Numbers k such that (4*k)!/(4*k + 3) is an integer. - Peter Bala, Jan 25 2017

Examples

			Distribution of the positive terms in the following triangular array:
  *;
  3,  *;
  *,  8,  *;
  6,  *, 15,  *;
  *, 13,  *, 24,  *;
  9,  *, 22,  *, 35,  *;
  *, 18,  *, 33,  *, 48,  *;
etc., where * marks the noninteger values of (2*h*k + k + h-1)/2 with h >= k >= 1. - _Vincenzo Librandi_, Apr 22 2014

Links

Crossrefs

Complement of A095278. Cf. also A045751, A014076, A153170, A153088, A153329, A153343.

Programs

  • Magma
    [n: n in [0..110] |not IsPrime(4*n+3)]; // Vincenzo Librandi, Apr 22 2014\
  • Maple
    for n from 0 to 100 do
if irem(factorial(4*n), 4*n+3) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017
  • Mathematica
    Select[Range[150],!PrimeQ[4#+3]&] (* Harvey P. Dale, Jul 04 2011 *)
  • PARI
    is(n)=!isprime(4*n+3) \\ Charles R Greathouse IV, Aug 01 2016

Formula

a(n) = (A091236(n) - 3)/4.

A153170 Numbers k such that 3*k + 2 is not prime.

Original entry on oeis.org

2, 4, 6, 8, 10, 11, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36, 38, 39, 40, 41, 42, 44, 46, 47, 48, 50, 51, 52, 53, 54, 56, 58, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 78, 80, 81, 82, 84, 86, 88, 90, 91, 92, 94, 95, 96, 98, 99, 100, 101, 102
Offset: 1

Views

Author

Vincenzo Librandi, Dec 20 2008

Keywords

Comments

Contains the positive even numbers (A005843) and the odd numbers of the form 2*A059324(.) + 1. - R. J. Mathar, Nov 27 2010
Numbers k such that (3*k)!/(3*k + 2) is an integer. - Peter Bala, Jan 25 2017

Examples

			Distribution of the odd terms in the following triangular array:
  *;
  *,   *;
  *,  11,   *;
  *,   *,   *,   *;
  *,   *,  25,   *,   *;
  *,  21,   *,   *,  47,   *;
  *,   *,   *,   *,   *,   *,   *;
  *,   *,  39,   *,   *,  73,   *,   *;
  *,  31,   *,   *,  69,   *,   *, 107,   *;
  *,   *,   *,   *,   *,   *,   *,   *,   *,   *;
  *,   *,  53,   *,   *,  99,   *,   *, 145,   *,   *;
  *,  41,   *,   *,  91,   *,   *, 141,   *,   *, 191,   *;
etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h - 1)/3 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

Links

Crossrefs

Cf. A024893, A014076, A045751, A095277, A153088, A153329, A153343.

Programs

  • Magma
    [n: n in [1..110] | not IsPrime(3*n + 2)]; // Vincenzo Librandi, Oct 11 2012
  • Maple
    for n from 0 to 100 do
if irem(factorial(3*n), 3*n+2) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017
  • Mathematica
    Select[Range[1, 200], !PrimeQ[3*# + 2] &] (* Vincenzo Librandi, Oct 11 2012 *)
  • PARI
    for(n=1,200,if(!isprime(3*n+2), print1(n,", "))) \\  Joerg Arndt, Nov 27 2010

Extensions

Edited by N. J. A. Sloane, Jun 23 2010

A153329 Numbers k such that 5*k + 1 is not prime.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 35, 37, 39, 40, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 60, 61, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 87
Offset: 1

Views

Author

Vincenzo Librandi, Dec 23 2008

Keywords

Comments

Numbers k such that (5*k)!/(5*k + 1) is an integer. - Peter Bala, Jan 25 2017

Examples

			Distribution of the even terms in the following triangular array:
   *;
   *,  *;
   4,  *,  *;
   *,  *,  *, 16;
   *,  *,  *,  *, 24;
   *,  *, 18,  *,  *,  *;
   *,  *,  *,  *,  *,  *,  *;
  10,  *,  *,  *,  *, 44,  *,  *;
   *,  *,  *, 34,  *,  *,  *,  *, 72;
   *,  *,  *,  *, 46,  *,  *,  *,  *, 88;
   *,  *, 32,  *,  *,  *,  *, 78,  *,  *,  *;
etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h)/5 with h >= k >= 1. - _Vincenzo Librandi_, Jan 17 2013

Links

Crossrefs

Cf. A024894, A014076, A153170, A045751, A095277, A153088, A153343.

Programs

  • Magma
    [n: n in [0..150] | not IsPrime(5*n + 1)]; // Vincenzo Librandi, Jan 12 2013
  • Maple
    for n from 0 to 100 do
if irem(factorial(5*n), 5*n+1) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017
  • Mathematica
    Select[Range[0, 200], !PrimeQ[5*# + 1]&] (* Vincenzo Librandi, Jan 12 2013 *)

Extensions

Erroneous comment deleted by N. J. A. Sloane, Jun 23 2010
0 added by Arkadiusz Wesolowski, Aug 03 2011

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

Views

Author

Vincenzo Librandi, Dec 24 2008

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Programs

  • Magma
    [n: n in [0..150] | not IsPrime(5*n + 4)]; // Vincenzo Librandi, Jan 12 2013
  • Maple
    # 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
  • Mathematica
    Select[Range[0, 200], !PrimeQ[5*# + 4]&] (* Vincenzo Librandi, Jan 12 2013 *)

Extensions

Erroneous comment deleted by N. J. A. Sloane, Jun 23 2010
0 added by Arkadiusz Wesolowski, Aug 03 2011

A153309 Numbers k such that 3*k + 1 is not prime.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 21, 23, 25, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 75, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 93, 95
Offset: 1

Views

Author

Vincenzo Librandi, Dec 23 2008

Keywords

Comments

Terms (except 0) can be written as 3xy +- (x + y) for x > 0, y > 0. - Ron R Spencer, Aug 01 2016
Apart from a(2) = 1 the sequence comprises those numbers k such that (3*k)!/(3*k + 1) is an integer. - Peter Bala, Jan 25 2017

Examples

			Distribution of the even terms in the following triangular array:
                        *;
                      *   8;
                    *   *  16;
                  *   *   *   *;
                *  18   *   *  40;
              *   *  30   *   *  56;
            *   *   *   *   *   *   *;
          *  28   *   *  62   *   *  96;
        *   *  44   *   *  82   *   *  120;
      *   *   *   *   *   *   *   *   *   *;
    *  38   *   *  84   *   *  130  *   *  176;
  *   *  58   *   *  108  *   *  158  *   *  208;
etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h)/3 with h >= k >= 1. - _Vincenzo Librandi_, Jan 17 2013

Links

Crossrefs

Cf. A024892, A014076, A153170, A045751, A095277, A153088, A153329, A153343.

Programs

  • Magma
    [n: n in [0..150] | not IsPrime(3*n + 1)]; // Vincenzo Librandi, Jan 12 2013
  • Maple
    # produces the sequence apart from the term equal to 1
for n from 0 to 100 do
if irem(factorial(3*n), 3*n+1) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017
  • Mathematica
    Select[Range[0, 200], !PrimeQ[3*# + 1]&] (* Vincenzo Librandi, Jan 12 2013 *)
  • PARI
    is(n)=!isprime(3*n+1) \\ Charles R Greathouse IV, Aug 01 2016

Extensions

Erroneous comment deleted by N. J. A. Sloane, Jun 23 2010
0 added by Arkadiusz Wesolowski, Jun 25 2011

A153877 Numbers n such that 5n-1 and 5n+1 are both prime.

Original entry on oeis.org

6, 12, 30, 36, 48, 54, 84, 114, 120, 132, 162, 204, 210, 246, 258, 264, 324, 390, 426, 462, 468, 510, 546, 558, 594, 600, 624, 660, 666, 672, 678, 708, 786, 804, 810, 846, 852, 930, 960, 1002, 1020, 1056, 1104, 1128, 1170, 1176, 1218, 1254, 1260, 1272
Offset: 1

Views

Author

Vincenzo Librandi, Jan 03 2009

Keywords

Comments

a(n) is an integer multiple of 6. - Alois P. Heinz, Jan 04 2009

Examples

			n=6, 5n-1=29, 5n+1=31; n=30, 5n-1=149, 5n+1=151.

Links

Crossrefs

Cf. A153355, A153088.

Programs

  • Magma
    [ n: n in [0..2000] | IsPrime(5*n-1) and IsPrime(5*n+1)]; // Vincenzo Librandi, Jul 31 2012
  • Maple
    a:= proc(n::posint) option remember; local m; if n=1 then 6 else for m from a(n-1)+6 by 6 while not (isprime(5*m-1) and isprime(5*m+1)) do od; m fi end: seq(a(n), n=1..100); # Alois P. Heinz, Jan 04 2009
  • Mathematica
    Select[Range[0,100001],PrimeQ[5#-1]&& PrimeQ[5#+1]&] (* Vincenzo Librandi, Jul 31 2012 *)

Extensions

Corrected and extended by Alois P. Heinz and R. J. Mathar, Jan 04 2009
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.