A045751 -id:A045751 - cp's OEIS frontend

A046954 Numbers k such that 6*k + 1 is nonprime.

0, 4, 8, 9, 14, 15, 19, 20, 22, 24, 28, 29, 31, 34, 36, 39, 41, 42, 43, 44, 48, 49, 50, 53, 54, 57, 59, 60, 64, 65, 67, 69, 71, 74, 75, 78, 79, 80, 82, 84, 85, 86, 88, 89, 92, 93, 94, 97, 98, 99, 104, 106, 108, 109, 111, 113, 114, 116, 117, 119, 120, 124, 127, 129, 130, 132, 133, 134, 136, 139, 140

Offset: 1

Felice Russo

nonn

Equals A171696 U A121763; A121765 U A171696 = A046953; A121763 U A121765 = A067611 where A067611 U A002822 U A171696 = A001477. - Juri-Stepan Gerasimov, Feb 13 2010, Feb 15 2010

These numbers (except 0) can be written as 6xy +-(x+y) for x > 0, y > 0. - Ron R Spencer, Aug 01 2016

			a(2)=8 because 6*8 + 1 = 49, which is composite.

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

Cf. A047845 (2n+1), A045751 (4n+1), A127260 (8n+1).

Cf. A046953, A008588, A016921, subsequence of A067611, complement of A024899.

Filtered([0..250], k-> not IsPrime(6*k+1)) # G. C. Greubel, Feb 21 2019

a046954 n = a046954_list !! (n-1)
a046954_list = map (`div` 6) $ filter ((== 0) . a010051' . (+ 1)) [0,6..]
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [0..250] | not IsPrime(6*n+1)]; // G. C. Greubel, Feb 21 2019

remove(k-> isprime(6*k+1), [$0..140])[]; # Muniru A Asiru, Feb 22 2019

a = Flatten[Table[If[PrimeQ[6*n + 1] == False, n, {}], {n, 0, 50}]] (* Roger L. Bagula, May 17 2007 *)
Select[Range[0, 200], !PrimeQ[6 # + 1] &] (* Vincenzo Librandi, Sep 27 2013 *)

is(n)=!isprime(6*n+1) \\ Charles R Greathouse IV, Aug 01 2016

[n for n in (0..250) if not is_prime(6*n+1)] # G. C. Greubel, Feb 21 2019

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar
Corrected by Juri-Stepan Gerasimov, Feb 13 2010, Feb 15 2010
Corrected by Vincenzo Librandi, Sep 27 2013

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

Antti Karttunen, Jun 01 2004

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

			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

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

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

[n: n in [0..110] |not IsPrime(4*n+3)]; // Vincenzo Librandi, Apr 22 2014\

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

Select[Range[150],!PrimeQ[4#+3]&] (* Harvey P. Dale, Jul 04 2011 *)

is(n)=!isprime(4*n+3) \\ Charles R Greathouse IV, Aug 01 2016

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

A153088 Numbers k such that 5*k - 1 is not prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 83, 85, 86, 87, 89, 91, 92, 93

Offset: 1

Vincenzo Librandi, Jan 03 2009

			Distribution of the even terms in the following triangular array:
   2;
   *,   *;
   *,   *,  10;
   *,   *,   *,   *;
   *,   *,   *,  20,   *;
   8,   *,   *,   *,   *,  34;
   *,   *,   *,   *,   *,   *,   *;
   *,   *,  24,   *,   *,   *,   *,  58;
   *,   *,   *,   *,  42,   *,   *,   *,   *;
   *,   *,   *,  38,   *,   *,   *,   *,  80,   *;
  14,   *,   *,   *,   *,  60,   *,   *,   *,   *, 106;
etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h + 2)/5 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

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

Cf. A153355, A014076, A153170, A045751, A095277, A153329, A153343.

[n: n in [1..100] | not IsPrime(5*n-1)]; // Vincenzo Librandi, Oct 11 2012

# produces the sequence apart from the initial terms 1 and 2
for n from 0 to 100 do
  if irem(factorial(5*n), 5*n+4) = 0 then print(n+1); end if;
end do: # Peter Bala, Jan 25 2017

Select[Range[1, 200], !PrimeQ[5 # - 1] &] (* Vincenzo Librandi, Oct 11 2012 *)

a(n) = A153343(n) + 1. - Peter Bala, Jan 25 2017

First 29 replaced with 20, 4 replaced with 44, extended by R. J. Mathar, Jan 05 2009

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

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

Vincenzo Librandi, Dec 20 2008

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

			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

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

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

[n: n in [1..110] | not IsPrime(3*n + 2)]; // Vincenzo Librandi, Oct 11 2012

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

Select[Range[1, 200], !PrimeQ[3*# + 2] &] (* Vincenzo Librandi, Oct 11 2012 *)

for(n=1,200,if(!isprime(3*n+2), print1(n,", "))) \\  Joerg Arndt, Nov 27 2010

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

Vincenzo Librandi, Dec 23 2008

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

			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

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

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

[n: n in [0..150] | not IsPrime(5*n + 1)]; // Vincenzo Librandi, Jan 12 2013

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

Select[Range[0, 200], !PrimeQ[5*# + 1]&] (* Vincenzo Librandi, Jan 12 2013 *)

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

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

A094897 If 4n+1 is not prime and 4n+3 is prime then a(n)=4*n+3, else a(n)=0.

Original entry on oeis.org

3, 0, 11, 0, 0, 23, 0, 0, 0, 0, 0, 47, 0, 0, 59, 0, 67, 71, 0, 79, 83, 0, 0, 0, 0, 0, 107, 0, 0, 0, 0, 127, 131, 0, 0, 0, 0, 0, 0, 0, 163, 167, 0, 0, 179, 0, 0, 191, 0, 0, 0, 0, 211, 0, 0, 223, 227, 0, 0, 239, 0, 0, 251, 0, 0, 263, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 307, 311, 0, 0, 0, 0, 331, 0

Offset: 0

Roger L. Bagula, Jun 14 2004

nonn

Cf. A045751, A095278, A094896.

[IsPrime(4*n+3) and not IsPrime(4*n+1) select 4*n+3 else 0:n in [0..85]]; // Marius A. Burtea, Nov 15 2019

A094897 := proc(n)
    if not isprime(4*n+1) and isprime(4*n+3) then
        4*n+3;
    else
        0;
    end if;
end proc:
seq(A094897(n),n=0..86) ; # R. J. Mathar, Nov 15 2019

a=Table[If[PrimeQ[4*n+1]==False&&PrimeQ[4*n+3]==True, 4*n+3, 0], {n, 0, 200}]

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

Vincenzo Librandi, Dec 23 2008

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

			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

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

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

[n: n in [0..150] | not IsPrime(3*n + 1)]; // Vincenzo Librandi, Jan 12 2013

# 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

Select[Range[0, 200], !PrimeQ[3*# + 1]&] (* Vincenzo Librandi, Jan 12 2013 *)

is(n)=!isprime(3*n+1) \\ Charles R Greathouse IV, Aug 01 2016

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

0 added by Arkadiusz Wesolowski, Jun 25 2011

A153085 Numbers k such that 4*k + 5 is not prime.

Original entry on oeis.org

1, 4, 5, 7, 10, 11, 13, 15, 16, 18, 19, 20, 22, 25, 28, 29, 30, 31, 32, 34, 35, 37, 39, 40, 41, 43, 45, 46, 49, 50, 51, 52, 53, 54, 55, 58, 60, 61, 62, 64, 65, 67, 70, 71, 73, 74, 75, 76, 79, 80, 81, 82, 84, 85, 88, 89, 90, 91, 93, 94, 95, 97, 100, 102, 103, 105, 106

Offset: 1

Vincenzo Librandi, Dec 18 2008

nonn

Let p=2n+1 be an odd number, then A140869(n,n) = (p^2-5)/4 = A028387(n-1).

One less than the associated entry in A045751: a(n) = A045751(n+1)-1. - R. J. Mathar, Jan 05 2011

			Triangle begins:
1;
*, 5;
4, *, 11;
*, 10, *, 19;
7, *, 18, *, 29;
*, 15, *, 28, *, 41;
where * mark the entries in A140869 which are non-integer if floor(.) is not applied there.

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

Cf. A028387, A045751, A111215, A140869.

[n: n in [1..120] | not IsPrime(4*n + 5)]; // Vincenzo Librandi, Oct 15 2012

Select[Range[200], !PrimeQ[4 # + 5] &] (* Vincenzo Librandi, Oct 15 2012 *)

A127260 Indices n of odd numbers of the form 8*n+1 that are not primes.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 8, 10, 13, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 31, 33, 34, 36, 37, 38, 40, 41, 43, 45, 46, 47, 48, 49, 52, 53, 55, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 73, 76, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 97, 98, 99, 100, 102

Offset: 1

Roger L. Bagula, May 17 2007

nonn

Could be generated from A045751 by removing all odd terms there, then halving each remaining term. - R. J. Mathar, Jun 08 2007

Cf. A045751.

isA127260 := proc(n) not isprime(8*n+1) ; end: for n from 0 to 200 do if isA127260(n) then printf("%d, ",n) ; fi ; od ; # R. J. Mathar, Jun 08 2007

a = Flatten[Table[If[PrimeQ[8*n + 1] == False, n, {}], {n, 0, 50}]]

More terms from R. J. Mathar, Jun 08 2007

cp's OEIS Frontend

A046954 Numbers k such that 6*k + 1 is nonprime.

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

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A153088 Numbers k such that 5*k - 1 is not prime.

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

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

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

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A094897 If 4n+1 is not prime and 4n+3 is prime then a(n)=4*n+3, else a(n)=0.