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.

A164926 Least prime p such that x^2+x+p produces primes for x=0..n-1 and composite for x=n.

Original entry on oeis.org

2, 3, 107, 5, 347, 1607, 1277, 21557, 51867197, 11, 180078317, 1761702947, 8776320587, 27649987598537, 291598227841757, 17
Offset: 1

Views

Author

T. D. Noe, Sep 01 2009

Keywords

Comments

Other known values: a(16)=17 and a(40)=41 (which is generated by Euler's polynomial, A005846). There are no other terms less than 10^12. All of Euler's Lucky numbers, A014556, are in this sequence. Assuming the prime k-tuples conjecture, Mollin's theorem 2.1 shows this sequence is defined for n>0.
a(21)=234505015943235329417 found by J. Waldvogel and Peter Leikauf. [Jens Kruse Andersen, Sep 09 2009]

Links

Crossrefs

Cf. A005846, A014556, A067774, A210360, A210361, A210362, A210363, A210364, A210365, A211236, A211237, A211238, A211239, A211240, A354585.

Programs

  • Mathematica
    PrimeRun[p_Integer] := Module[{k=0}, While[PrimeQ[k^2+k+p], k++ ]; k]; nn=8; t=Table[0,{nn}]; cnt=0; p=1; While[cnt

Extensions

a(14) and a(15) from Jens Kruse Andersen, Sep 09 2009

A210361 Prime numbers p such that x^2 + x + p produces primes for x = 0..2 but not x = 3.

Original entry on oeis.org

107, 191, 311, 461, 821, 857, 881, 1301, 1871, 1997, 2081, 2237, 2267, 2657, 3251, 3461, 3671, 4517, 4967, 5231, 5477, 5501, 5651, 6197, 6827, 7877, 8087, 8291, 8537, 8861, 9431, 10427, 10457, 11171, 12917, 13001, 13691, 13757, 13877, 14081, 14321, 15641
Offset: 1

Views

Author

T. D. Noe, Apr 05 2012

Keywords

Comments

The first term is A164926(3).

Links

Crossrefs

Cf. A067774, A164926, A210360, A210362, A210363, A210364, A210365.

Programs

  • Mathematica
    lookfor = 3; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t

A210362 Prime numbers p such that x^2 + x + p produces primes for x = 0..3 but not x = 4.

Original entry on oeis.org

5, 101, 227, 1091, 1481, 1487, 3917, 4127, 4787, 8231, 9461, 10331, 11777, 12107, 14627, 16061, 20747, 25577, 27737, 29021, 32297, 33347, 35531, 35591, 36467, 38447, 39227, 41177, 42461, 44267, 44531, 49031, 59441, 69191, 77237, 79811, 80777, 93251, 93491
Offset: 1

Views

Author

T. D. Noe, Apr 05 2012

Keywords

Comments

The first term is A164926(4).

Links

Crossrefs

Cf. A067774, A164926, A210360, A210361, A210363, A210364, A210365.

Programs

  • Mathematica
    lookfor = 4; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t
Select[Prime[Range[10000]],AllTrue[#+{2,6,12},PrimeQ]&&!PrimeQ[#+20]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 26 2015 *)
Select[Prime[Range[10000]],Boole[PrimeQ[#+{2,6,12,20}]]=={1,1,1,0}&] (* Harvey P. Dale, Nov 17 2024 *)

A210363 Prime numbers p such that x^2 + x + p produces primes for x = 0..4 but not x = 5.

Original entry on oeis.org

347, 641, 1427, 2687, 4001, 4637, 4931, 19421, 21011, 22271, 23741, 26711, 27941, 32057, 43781, 45821, 55331, 55817, 68207, 71327, 91571, 128657, 165701, 167621, 172421, 179897, 191447, 193871, 205421, 234191, 239231, 258107, 258611, 259157, 278807, 290021
Offset: 1

Views

Author

T. D. Noe, Apr 05 2012

Keywords

Comments

The first term is A164926(5).

Links

Crossrefs

Cf. A067774, A164926, A210360, A210361, A210362, A210364, A210365.

Programs

  • Mathematica
    lookfor = 5; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t
Select[Prime[Range[26000]],AllTrue[#+{2,6,12,20},PrimeQ] && !PrimeQ[ #+30]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 13 2017 *)
Select[Prime[Range[26000]],Boole[PrimeQ[#+{2,6,12,20,30}]]=={1,1,1,1,0}&] (* Harvey P. Dale, May 29 2025 *)

A210364 Prime numbers p such that x^2 + x + p produces primes for x = 0..5 but not x = 6.

Original entry on oeis.org

1607, 3527, 13901, 31247, 33617, 55661, 68897, 97367, 166841, 195731, 221717, 347981, 348431, 354371, 416387, 506327, 548831, 566537, 929057, 954257, 1246367, 1265081, 1358801, 1505087, 1538081, 1595051, 1634441, 1749257, 2200811, 2322107, 2641547, 2697971
Offset: 1

Views

Author

T. D. Noe, Apr 05 2012

Keywords

Comments

The first term is A164926(6).

Links

Crossrefs

Cf. A067774, A164926, A210360, A210361, A210362, A210363, A210365.

Programs

  • Mathematica
    lookfor = 6; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t

A210365 Prime numbers p such that x^2 + x + p produces primes for x = 0..6 but not x = 7.

Original entry on oeis.org

1277, 28277, 113147, 421697, 665111, 1164587, 1615631, 2798921, 2846771, 3053747, 5071667, 5093507, 5344247, 5706641, 6383051, 8396777, 10732817, 10812407, 11920367, 13176587, 16197947, 16462541, 16655447, 16943471, 17807831, 18102101, 20488901, 23421311
Offset: 1

Views

Author

T. D. Noe, Apr 05 2012

Keywords

Comments

The first term is A164926(7).

Links

Crossrefs

Cf. A067774, A164926, A210360, A210361, A210362, A210363, A210364.

Programs

  • Mathematica
    lookfor = 7; t = {}; n = 0; While[Length[t] < 30, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t

A297709 Table read by antidiagonals: Let b be the number of digits in the binary expansion of n. Then T(n,k) is the k-th odd prime p such that the binary digits of n match the primality of the b consecutive odd numbers beginning with p (or 0 if no such k-th prime exists).

Original entry on oeis.org

3, 5, 7, 7, 13, 3, 11, 19, 5, 23, 13, 23, 11, 31, 7, 17, 31, 17, 47, 13, 5, 19, 37, 29, 53, 19, 11, 3, 23, 43, 41, 61, 37, 17, 0, 89, 29, 47, 59, 73, 43, 29, 0, 113, 23, 31, 53, 71, 83, 67, 41, 0, 139, 31, 19, 37, 61, 101, 89, 79, 59, 0, 181, 47, 43, 7, 41, 67
Offset: 1

Views

Author

Jon E. Schoenfield, Apr 15 2018

Keywords

Comments

For each n >= 1, row n is the union of rows 2n and 2n+1.
Rows with no nonzero terms: 15, 21, 23, 28, 30, 31, ...
Rows whose only nonzero term is 3: 7, 14, 29, 59, 118, 237, 475, 950, 1901, 3802, 7604, ...
Rows whose only nonzero term is 5: 219, 438, 877, 1754, 3508, 7017, 14035, ...
For j = 2, 3, 4, ..., respectively, the first row whose only nonzero term is prime(j) is 7, 219, 2921, ...; is there such a row for every odd prime?

Examples

			13 = 1101_2, so row n=13 lists the odd primes p such that the four consecutive odd numbers p, p+2, p+4, and p+6 are prime, prime, composite, and prime, respectively; these are the terms of A022004.
14 = 1110_2, so row n=14 lists the odd primes p such that p, p+2, p+4, and p+6 are prime, prime, prime, and composite, respectively; since there is only one such prime (namely, 3), there is no such 2nd, 3rd, 4th, etc. prime, so the terms in row 14 are {3, 0, 0, 0, ...}.
15 = 1111_2, so row n=15 would list the odd primes p such that p, p+2, p+4, and p+6 are all prime, but since no such prime exists, every term in row 15 is 0.
Table begins:
  n in base|                    k                   |  OEIS
  ---------+----------------------------------------+sequence
  10     2 |   1    2    3    4    5    6    7    8 | number
  =========+========================================+========
   1     1 |   3    5    7   11   13   17   19   23 | A065091
   2    10 |   7   13   19   23   31   37   43   47 | A049591
   3    11 |   3    5   11   17   29   41   59   71 | A001359
   4   100 |  23   31   47   53   61   73   83   89 | A124582
   5   101 |   7   13   19   37   43   67   79   97 | A029710
   6   110 |   5   11   17   29   41   59   71  101 | A001359*
   7   111 |   3    0    0    0    0    0    0    0 |
   8  1000 |  89  113  139  181  199  211  241  283 | A083371
   9  1001 |  23   31   47   53   61   73   83  131 | A031924
  10  1010 |  19   43   79  109  127  163  229  313 |
  11  1011 |   7   13   37   67   97  103  193  223 | A022005
  12  1100 |  29   59   71  137  149  179  197  239 | A210360*
  13  1101 |   5   11   17   41  101  107  191  227 | A022004
  14  1110 |   3    0    0    0    0    0    0    0 |
  15  1111 |   0    0    0    0    0    0    0    0 |
  16 10000 | 113  139  181  199  211  241  283  293 | A124584
  17 10001 |  89  359  389  401  449  479  491  683 | A031926
  18 10010 |  31   47   61   73   83  151  157  167 |
  19 10011 |  23   53  131  173  233  263  563  593 | A049438
  20 10100 |  19   43   79  109  127  163  229  313 |
  21 10101 |   0    0    0    0    0    0    0    0 |
  22 10110 |   7   13   37   67   97  103  193  223 | A022005
  23 10111 |   0    0    0    0    0    0    0    0 |
  24 11000 | 137  179  197  239  281  419  521  617 |
  25 11001 |  29   59   71  149  269  431  569  599 | A049437*
  26 11010 |  17   41  107  227  311  347  461  641 |
  27 11011 |   5   11  101  191  821 1481 1871 2081 | A007530
  28 11100 |   0    0    0    0    0    0    0    0 |
  29 11101 |   3    0    0    0    0    0    0    0 |
  30 11110 |   0    0    0    0    0    0    0    0 |
  31 11111 |   0    0    0    0    0    0    0    0 |
*other than the referenced sequence's initial term 3
.
Alternative version of table:
.
  n in base|primal-|               k              |  OEIS
  ---------+  ity  +------------------------------+  seq.
  10     2 |pattern|   1    2    3    4    5    6 | number
  =========+=======+==============================+========
   1     1 | p     |   3    5    7   11   13   17 | A065091
   2    10 | pc    |   7   13   19   23   31   37 | A049591
   3    11 | pp    |   3    5   11   17   29   41 | A001359
   4   100 | pcc   |  23   31   47   53   61   73 | A124582
   5   101 | pcp   |   7   13   19   37   43   67 | A029710
   6   110 | ppc   |   5   11   17   29   41   59 | A001359*
   7   111 | ppp   |   3    0    0    0    0    0 |
   8  1000 | pccc  |  89  113  139  181  199  211 | A083371
   9  1001 | pccp  |  23   31   47   53   61   73 | A031924
  10  1010 | pcpc  |  19   43   79  109  127  163 |
  11  1011 | pcpp  |   7   13   37   67   97  103 | A022005
  12  1100 | ppcc  |  29   59   71  137  149  179 | A210360*
  13  1101 | ppcp  |   5   11   17   41  101  107 | A022004
  14  1110 | pppc  |   3    0    0    0    0    0 |
  15  1111 | pppp  |   0    0    0    0    0    0 |
  16 10000 | pcccc | 113  139  181  199  211  241 | A124584
  17 10001 | pcccp |  89  359  389  401  449  479 | A031926
  18 10010 | pccpc |  31   47   61   73   83  151 |
  19 10011 | pccpp |  23   53  131  173  233  263 | A049438
  20 10100 | pcpcc |  19   43   79  109  127  163 |
  21 10101 | pcpcp |   0    0    0    0    0    0 |
  22 10110 | pcppc |   7   13   37   67   97  103 | A022005
  23 10111 | pcppp |   0    0    0    0    0    0 |
  24 11000 | ppccc | 137  179  197  239  281  419 |
  25 11001 | ppccp |  29   59   71  149  269  431 | A049437*
  26 11010 | ppcpc |  17   41  107  227  311  347 |
  27 11011 | ppcpp |   5   11  101  191  821 1481 | A007530
  28 11100 | pppcc |   0    0    0    0    0    0 |
  29 11101 | pppcp |   3    0    0    0    0    0 |
  30 11110 | ppppc |   0    0    0    0    0    0 |
  31 11111 | ppppp |   0    0    0    0    0    0 |
.
     *other than the referenced sequence's initial term 3

Crossrefs

Cf. A001359, A007530, A022004, A022005, A029710, A031924, A031926, A049437, A049438, A049591, A065091, A124582, A083371, A124584, A210360.
Showing 1-7 of 7 results.

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

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