The On-Line Encyclopedia of Integer Sequences (OEIS)

A297709 revision #28

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).

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

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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 | 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.

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