A260717 Square array: row n gives the numbers remaining before the stage n of Ludic sieve.

2, 3, 3, 4, 5, 5, 5, 7, 7, 7, 6, 9, 11, 11, 11, 7, 11, 13, 13, 13, 13, 8, 13, 17, 17, 17, 17, 17, 9, 15, 19, 23, 23, 23, 23, 23, 10, 17, 23, 25, 25, 25, 25, 25, 25, 11, 19, 25, 29, 29, 29, 29, 29, 29, 29, 12, 21, 29, 31, 37, 37, 37, 37, 37, 37, 37, 13, 23, 31, 37, 41, 41, 41, 41, 41, 41, 41, 41, 14, 25, 35, 41, 43, 43, 43, 43, 43, 43, 43, 43, 43

Offset: 1

Author

Antti Karttunen, Jul 30 2015

Keywords

• nonn
• tabl

Comments

This square array A(row,col) is read by downwards antidiagonals as: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Ludic sieve starts with natural numbers larger than one: 2, 3, 4, 5, 6, 7, ... and in each subsequent stage one sets k = (which will be one of Ludic numbers) and removes both k and every k-th term after it, from column positions 1, 1+k, 1+2k, 1+3k, etc. of the preceding row to produce the next row of this array.

Examples

			The top left corner of the array:
   2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  14,  15,  16,  17
   3,  5,  7,  9, 11, 13, 15, 17,  19,  21,  23,  25,  27,  29,  31,  33
   5,  7, 11, 13, 17, 19, 23, 25,  29,  31,  35,  37,  41,  43,  47,  49
   7, 11, 13, 17, 23, 25, 29, 31,  37,  41,  43,  47,  53,  55,  59,  61
  11, 13, 17, 23, 25, 29, 37, 41,  43,  47,  53,  55,  61,  67,  71,  73
  13, 17, 23, 25, 29, 37, 41, 43,  47,  53,  61,  67,  71,  73,  77,  83
  17, 23, 25, 29, 37, 41, 43, 47,  53,  61,  67,  71,  77,  83,  89,  91
  23, 25, 29, 37, 41, 43, 47, 53,  61,  67,  71,  77,  83,  89,  91,  97
  25, 29, 37, 41, 43, 47, 53, 61,  67,  71,  77,  83,  89,  91,  97, 107
  29, 37, 41, 43, 47, 53, 61, 67,  71,  77,  83,  89,  91,  97, 107, 115
  37, 41, 43, 47, 53, 61, 67, 71,  77,  83,  89,  91,  97, 107, 115, 119
  41, 43, 47, 53, 61, 67, 71, 77,  83,  89,  91,  97, 107, 115, 119, 121
  43, 47, 53, 61, 67, 71, 77, 83,  89,  91,  97, 107, 115, 119, 121, 127
  47, 53, 61, 67, 71, 77, 83, 89,  91,  97, 107, 115, 119, 121, 127, 131
  53, 61, 67, 71, 77, 83, 89, 91,  97, 107, 115, 119, 121, 127, 131, 143
  61, 67, 71, 77, 83, 89, 91, 97, 107, 115, 119, 121, 127, 131, 143, 149
  etc.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array
• Index entries for sequences generated by sieves

Crossrefs

Transpose: A260718.

Column 1: A003309 (without the initial 1).

Row 1: A020725, Row 2: A144396, Row 3: A007310 (from its second term onward), Row 4: A260714, Row 5: A260715.

Cf. A255127 (gives the numbers removed at each stage).

Cf. also array A258207.

Programs

Scheme

(define (A260717 n) (A260717bi (A002260 n) (A004736 n)))
(define (A260717bi row col) ((rowfun_n_for_A003309sieve row) col))
(define (add1 n) (1+ n))
;; Uses definec-macro which can memoize also function-closures:
(definec (rowfun_n_for_A003309sieve n) (if (= 1 n) add1 (let* ((prevrowfun (rowfun_n_for_A003309sieve (- n 1))) (everynth (prevrowfun 1))) (compose-funs prevrowfun (nonzero-pos 1 1 (lambda (i) (modulo (- i 1) everynth)))))))