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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). 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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?

LINKS

Table of n, a(n) for n=1..68.

EXAMPLE

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 | A124583

   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 | A124583

   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, A124583, A124584, A210360.

Sequence in context: A335046 A060265 A172365 * A242999 A098566 A006540

Adjacent sequences:  A297706 A297707 A297708 * A297710 A297711 A297712

KEYWORD

nonn,tabl

AUTHOR

Jon E. Schoenfield, Apr 15 2018

STATUS

approved

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Last modified December 9 07:21 EST 2021. Contains 349627 sequences. (Running on oeis4.)