A173028 Partition of the row numbers of the Wythoff array W: two numbers are in the same row if and only if their rows in W have (essentially) a common divisor greater than 1.
1, 3, 2, 4, 9, 6, 5, 13, 29, 7, 16, 45, 43, 35, 8, 19, 56, 57, 52, 15, 10, 22, 67, 186, 181, 58, 51, 11, 25, 78, 223, 226, 77, 199, 55, 12, 28, 89, 260, 271, 96, 265, 82, 61, 14, 31, 262, 297, 316, 115, 331, 109, 91, 71, 17, 34, 291, 334, 361, 351, 397, 136, 317, 106, 87, 18
Offset: 1
Examples
First four rows of R: 1...3....4....5.....16....19....22...25...28... 2...9....13...45....56....67....78...89...262.. 6...29...43...57....186...223...260..297..334... 7...35...52...181...226...271...316..361..1063... For example, row 3 begins with 6, which is the least positive integer not in rows 1 and 2. Row 6 of W is (14,23,37,60,...) Row 29 of W is (74,120,194,...) = 2*(37,60,97...). Row 43 of W is (111,180,291,...) = 3*(37,60,97,...). So row 3 of R begins with (6,29,43...) as there are no other rows of W numbered <43 which are multiples of row 6 of W.
Formula
Let R(n,k) be the number in row n, column k. After Row 1 (A173027),
inductively, R(n,1) is the least positive integer not in the first n-1
rows, and the rest of row n consists of the numbers of rows X of the
Wythoff array W for X a multiple of a tail of row R(n,1) of W.
Extensions
Corrections (these have been made): a(31) should read 223 instead of 225, a(63) 317 instead of 314 - K. G. Stier, Dec 21 2012
Comments