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

1,1

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 = <the initial term of the preceding row> (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.

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

Example

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.

Prog

(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)))))))

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.

Sequence in context: A088023 A324477 A287292 * A282719 A061451 A205542

Adjacent sequences: A260714 A260715 A260716 * A260718 A260719 A260720

Keyword

nonn,tabl

Author

Antti Karttunen, Jul 30 2015

Status

approved