|
| |
|
|
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
|
|
|
|
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
=========+=======+==============================+========
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.
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
