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 1-10 of 10 results.

A138685 Numbers k such that there is no prime of the form 2k + p^2 for any prime p.

Original entry on oeis.org

13, 28, 34, 43, 55, 58, 67, 73, 76, 88, 97, 100, 103, 106, 118, 133, 139, 145, 148, 157, 160, 163, 166, 178, 181, 184, 193, 199, 202, 208, 214, 223, 232, 238, 244, 253, 259, 262, 265, 268, 271, 283, 286, 298, 301, 307, 310, 313, 328, 331, 340, 343, 349, 352
Offset: 1

Views

Author

Artur Jasinski, Mar 26 2008

Keywords

Comments

Indices where zero occurs in A138479.
For primes in this sequences see A138686.

Links

Crossrefs

Cf. A138479, A138686.

Programs

  • Mathematica
    a = {}; Do[p = 0; While[(! PrimeQ[2*n + Prime[p + 1]^2]) && (p < 10000), p++ ]; If[p < 10000,[null], AppendTo[a, n]], {n, 1, 550}]; a
Select[Range[400],Mod[#,3]==1&&CompositeQ[2#+9]&] (* Harvey P. Dale, Feb 23 2017 *)

Formula

Based on comments from Zak Seidov, Don Reble and M. F. Hasler, we now know that these are the numbers k such that k == 1 (mod 3) and 2k + 9 is composite. - N. J. A. Sloane, Apr 20 2008

A138686 Primes in A138685.

Original entry on oeis.org

13, 43, 67, 73, 97, 103, 139, 157, 163, 181, 193, 199, 223, 271, 283, 307, 313, 331, 349, 373, 379, 397, 421, 433, 457, 463, 499, 523, 601, 613, 619, 631, 643, 661, 673, 691, 727, 733, 769, 811, 823, 853, 859, 877, 883, 937, 967, 991, 1021, 1033, 1063, 1069
Offset: 1

Views

Author

Artur Jasinski, Mar 26 2008

Keywords

Links

Crossrefs

Cf. A138479, A138685, A138687.

A138694 Numbers n such that the set {2*n+p^2, p any prime} contains exactly one prime.

Original entry on oeis.org

1, 4, 7, 10, 16, 19, 22, 25, 31, 37, 40, 46, 49, 52, 61, 64, 70, 79, 82, 85, 91, 94, 109, 112, 115, 121, 124, 127, 130, 136, 142, 151, 154, 169, 172, 175, 187, 190, 196, 205, 211, 217, 220, 226, 229, 235, 241, 247, 250, 256, 274, 277, 280, 289, 292, 295, 304, 316
Offset: 1

Views

Author

Artur Jasinski, Mar 27 2008

Keywords

Comments

The sequence forms a subset of A016777, as explained below:
For each prime p<>3 we have p^2 =1 (mod 3), see A024700.
(i) For the k where 2*k=2 (mod 3), that is where k=1 (mod 3), this leads to 2*k+p^2=0 (mod 3), so the 2*k+p^2 are divisible by 3 (not prime) unless p=3.
The subcase where 2*k+3^2 is prime generates this sequence here; the subcase where it is not generates A138685.
(ii) For the k where 2*k=0 (mod 3), that is where k=0 (mod 3), one can select any p^2 =1 (mod 3)
to generate a prime 2*k+p^2 = 1 (mod 3), so these k generate many primes (of the form A002476).
(iii) For the k where 2*k=1 (mod 3), that is where k=2 (mod 3), one can select any p^2 =1 (mod 3)
to generate a prime 2*k+p^2 = 2 (mod 3), so these k generate many primes (of the form A003627).
The unique primes associated with each n are in A007528: n=1 associated with A007528(2)=11=2*1+3^2,
n=4 associated with A007528(3)=17=2*4+3^2 etc.

Examples

			3 is not in the sequence because {6+2^2, 6+3^3, 6+5^2, 6+7^2,..} = {10, 15, 31, 55,..,127,..,367,..}
contains the primes 31, 127, 367,..., generated with p=5,11,19...
4 is in the sequence because {8+2^2, 8+3^3, 8+5^2, 8+7^2,..} = {12, 17, 33, 57,...} contains
only one prime (that is, 17), generated with p=3.

Links

Crossrefs

Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

Programs

  • Mathematica
    b = {}; Do[a = {}; Do[If[PrimeQ[2*k + Prime[n]^2], AppendTo[a, k]], {n, 1, 100}]; If[Length[a] < 2, AppendTo[b, a]], {k, 1, 500}]; Union[Flatten[b]]

Formula

{This sequence here} Union {A138685} = {A016777}.

Extensions

Edited by R. J. Mathar, May 15 2009

A059324 Numbers n such that 6n + 5 is composite.

Original entry on oeis.org

5, 10, 12, 15, 19, 20, 23, 25, 26, 30, 33, 34, 35, 36, 40, 45, 47, 49, 50, 53, 54, 55, 56, 60, 61, 62, 65, 67, 68, 70, 72, 75, 78, 80, 82, 85, 87, 88, 89, 90, 91, 95, 96, 100, 101, 103, 104, 105, 110, 111, 114, 115, 117, 118, 120, 121, 122, 124, 125, 127, 129, 130
Offset: 1

Views

Author

Anton Joha, Jan 26 2001

Keywords

Comments

Conjecture: There exists no pair of primes (p, q > p^2) such that q - p^2 = 6*n - 4 (see A138479). - Philippe LALLOUET (philip.lallouet(AT)orange.fr), Mar 20 2008

Examples

			a(3) = 12 because 6*12 + 5 = 77 is composite.

Links

Crossrefs

Complement of A059325.
Cf. A138479.

Programs

  • Mathematica
    Select[Range[200],!PrimeQ[6#+5]&]  (* Harvey P. Dale, Mar 13 2011 *)
  • PARI
    isok(n) = ! isprime(6*n+5); \\ Michel Marcus, Jan 06 2017

Formula

a(n) = A046953(n-1) - 1.

Extensions

More terms from Henry Bottomley, Jan 29 2001

A138691 Numbers of the form 68+p^2 (where p is a prime).

Original entry on oeis.org

72, 77, 93, 117, 189, 237, 357, 429, 597, 909, 1029, 1437, 1749, 1917, 2277, 2877, 3549, 3789, 4557, 5109, 5397, 6309, 6957, 7989, 9477, 10269, 10677, 11517, 11949, 12837, 16197, 17229, 18837, 19389, 22269, 22869, 24717, 26637, 27957, 29997
Offset: 1

Views

Author

Artur Jasinski, Mar 26 2008

Keywords

Comments

68 + p^2 is divisible by 3 for any prime p > 3. - M. F. Hasler

Links

Crossrefs

Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

Programs

  • Mathematica
    Table[2*34 + Prime[p + 1]^2, {p, 0, 100}]
Prime[Range[50]]^2+68 (* Harvey P. Dale, Oct 19 2011 *)
  • PARI
    forprime(p=1, 1e2, print1(68+p^2, ", ")) \\ Felix Fröhlich, Jul 07 2014

A138692 Numbers of the form 86+p^2 (where p is a prime).

Original entry on oeis.org

90, 95, 111, 135, 207, 255, 375, 447, 615, 927, 1047, 1455, 1767, 1935, 2295, 2895, 3567, 3807, 4575, 5127, 5415, 6327, 6975, 8007, 9495, 10287, 10695, 11535, 11967, 12855, 16215, 17247, 18855, 19407, 22287, 22887, 24735, 26655, 27975, 30015
Offset: 1

Views

Author

Artur Jasinski, Mar 26 2008

Keywords

Comments

86+p^2 is divisible by 3, for any prime p>3. - M. F. Hasler

Crossrefs

Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

Programs

  • Mathematica
    Table[2*43 + Prime[p + 1]^2, {p, 0, 100}]
Prime[Range[40]]^2+86 (* Harvey P. Dale, Jan 31 2020 *)

A138693 Numbers of the form 110 + p^2. (where p is a prime).

Original entry on oeis.org

114, 119, 135, 159, 231, 279, 399, 471, 639, 951, 1071, 1479, 1791, 1959, 2319, 2919, 3591, 3831, 4599, 5151, 5439, 6351, 6999, 8031, 9519, 10311, 10719, 11559, 11991, 12879, 16239, 17271, 18879, 19431, 22311, 22911, 24759, 26679, 27999, 30039
Offset: 1

Views

Author

Artur Jasinski, Mar 26 2008

Keywords

Comments

110+p^2 is divisible by 3, for any prime p>3. - M. F. Hasler

Crossrefs

Cf. A000040, A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

Programs

Formula

a(n) = 110 + A000040(n)^2. - Wesley Ivan Hurt, Sep 13 2014
a(n) = 110 + A001248(n). - Michel Marcus, Sep 14 2014

A138687 Composite n with no prime of the form 2n + p^2 for any prime p.

Original entry on oeis.org

28, 34, 55, 58, 76, 88, 100, 106, 118, 133, 145, 148, 160, 166, 178, 184, 202, 208, 214, 232, 238, 244, 253, 259, 262, 265, 268, 286, 298, 301, 310, 328, 340, 343, 352, 358, 361, 364, 370, 385, 388, 391, 403, 412, 418, 430, 442, 445, 448, 454, 475, 478, 490
Offset: 1

Views

Author

Artur Jasinski, Mar 26 2008

Keywords

Crossrefs

Cf. A138479, A138685, A138686.

Formula

Composite members of A138685.

Extensions

Typo corrected by Michel Marcus, Nov 04 2013

A138689 Numbers of the form 26+p^2 (where p is a prime).

Original entry on oeis.org

30, 35, 51, 75, 147, 195, 315, 387, 555, 867, 987, 1395, 1707, 1875, 2235, 2835, 3507, 3747, 4515, 5067, 5355, 6267, 6915, 7947, 9435, 10227, 10635, 11475, 11907, 12795, 16155, 17187, 18795, 19347, 22227, 22827, 24675, 26595, 27915, 29955, 32067
Offset: 1

Views

Author

Artur Jasinski, Mar 26 2008

Keywords

Comments

26+p^2 is divisible by 3 for any prime p>3; Zak Seidov

Crossrefs

Cf. A138479, A138685, A138686.

Programs

  • Mathematica
    Table[2*13 + Prime[p + 1]^2, {p, 0, 100}]

A138690 Numbers of the form 56+p^2 (where p is a prime).

Original entry on oeis.org

60, 65, 81, 105, 177, 225, 345, 417, 585, 897, 1017, 1425, 1737, 1905, 2265, 2865, 3537, 3777, 4545, 5097, 5385, 6297, 6945, 7977, 9465, 10257, 10665, 11505, 11937, 12825, 16185, 17217, 18825, 19377, 22257, 22857, 24705, 26625, 27945, 29985
Offset: 1

Views

Author

Artur Jasinski, Mar 26 2008

Keywords

Comments

56+p^2 is divisible by 3 for any prime p>3. - M. F. Hasler

Crossrefs

Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

Programs

  • Mathematica
    Table[2*28 + Prime[p + 1]^2, {p, 0, 100}]
Prime[Range[40]]^2+56 (* Harvey P. Dale, Dec 15 2015 *)
Showing 1-10 of 10 results.

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