|
| |
|
|
A097858
|
|
Triangle read by rows where the top row is a 1. The m-th row of the triangle contains the smallest m as-yet-unused positive integers, in order from left to right, which are each coprime with every integer in the (m-1)th row.
|
|
1
|
|
|
|
1, 2, 3, 5, 7, 11, 4, 6, 8, 9, 13, 17, 19, 23, 25, 12, 14, 16, 18, 21, 22, 29, 31, 37, 41, 43, 47, 53, 10, 15, 20, 24, 26, 27, 28, 30, 59, 61, 67, 71, 73, 79, 83, 89, 97, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 60, 62, 63, 64, 65
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Comments from Isabel Lugo (izzycat(AT)gmail.com), Aug 31 2004:
"What seems to happen is that after not too long, the triangle settles down into alternating rows of prime and composite numbers, starting with:
"row #9: 59 61 67 71 73 79 83 89 97
"row #10: 32 33 34 35 36 38 39 40 42 44
"and basically, row 2n has enough small prime factors among its numbers that row 2n+1 is all primes because any composites have been filtered out.
"Then since row 2n+1 is all primes and the primes are less dense than the composites, there are unused composites that are smaller than any of the primes in row 2n+1 and these become row 2n+2."
For rows 7 through 32, the even rows contain the least numbers not already used (which are all composite) and the odd rows contain the least primes not already used. Row 31 has the primes from 1429 through 1619 and row 32 has the composites from 303 through 340. Since row 32 contains no multiple of 43, 43^2 = 1849 occurs in row 33. After that the even rows remain all composite and the odd rows remain mostly prime, but exceptions become more frequent. Note that when a small prime factor occurs in row 2k+1, it can't occur in row 2k+2, so this allows it to recur in row 2k+3. For example, rows 49, 51, 53, 55, 57 and 59 all contain multiples of 67. - David Wasserman, Dec 28 2007
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Triangle begins:
1
2 3
5 7 11
4 6 8 9
13 17 19 23 25
12 14 16 18 21 22
....
|
|
|
MATHEMATICA
|
a = row = {1}; Do[row = Take[Select[Complement[Range[Prime[PrimePi@Max@a + n]], a], GCD[#, Times @@ row] == 1 &], n]; a = Join[a, row], {n, 2, 14}]; a (* Ivan Neretin, May 23 2018 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
