A258207 Square array: row n gives the numbers remaining after the stage n of Lucky sieve.

1, 3, 1, 5, 3, 1, 7, 7, 3, 1, 9, 9, 7, 3, 1, 11, 13, 9, 7, 3, 1, 13, 15, 13, 9, 7, 3, 1, 15, 19, 15, 13, 9, 7, 3, 1, 17, 21, 21, 15, 13, 9, 7, 3, 1, 19, 25, 25, 21, 15, 13, 9, 7, 3, 1, 21, 27, 27, 25, 21, 15, 13, 9, 7, 3, 1, 23, 31, 31, 31, 25, 21, 15, 13, 9, 7, 3, 1, 25, 33, 33, 33, 31, 25, 21, 15, 13, 9, 7, 3, 1, 27, 37, 37, 37, 33, 31, 25, 21, 15, 13, 9, 7, 3, 1, 29, 39, 43, 43, 37, 33, 31, 25, 21, 15, 13, 9, 7, 3, 1

Offset: 1

Author

Antti Karttunen, Jul 27 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.

Lucky sieve starts with natural numbers: 1, 2, 3, 4, 5, 6, 7, ... from which at first stage the even numbers are removed, and on each subsequent stage n (n > 1) one sets k = (these k will form the Lucky numbers) and removes every k-th term (from column positions k, 2k, 3k, etc.) of the preceding row to produce the next row of this array.

On each row n, the first term that differs from the n-th Lucky number (A000959(n)) occurs at the column position A000959(n+1) and that number is A219178(n) when n > 1.

Examples

			The top left corner of the array:
1, 3, 5, 7,  9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39
1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49, 51, 55, 57
1, 3, 7, 9, 13, 15, 21, 25, 27, 31, 33, 37, 43, 45, 49, 51, 55, 57, 63, 67
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 45, 49, 51, 55, 63, 67, 69, 73
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 55, 63, 67, 69, 73, 75
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79
...
To get row 2 from row 1, we use the second term of the first row, which is 3, to remove every third term from row 1: 5, 11, 17, ... which leaves 1, 3, 7, 9, 13, ...
To get row 3 from row 2, we use the third term of row 2, which is 7, to remove every seventh term from row 2: 19, 39, ... which then results in the third row.
		

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

Cf. A000959 (Lucky numbers), which occur at the main and also any subdiagonal of this array. Also the rows converge towards A000959.

Row 1: A005408. Row 2: A047241. Row 3: A258011.

Transpose: A258208.

Cf. also A219178, A255543, A260717.

Programs

Scheme

(define (A258207 n) (A258207bi (A002260 n) (A004736 n)))
(define (A258207bi row col) ((rowfun_n_for_A000959sieve row) col))
;; Uses definec-macro which can memoize also function-closures:
(definec (rowfun_n_for_A000959sieve n) (if (= 1 n) A005408shifted (let* ((prevrowfun (rowfun_n_for_A000959sieve (- n 1))) (everynth (prevrowfun n))) (compose-funs prevrowfun (nonzero-pos 1 1 (lambda (i) (modulo i everynth)))))))
(define (A005408shifted n) (- (* 2 n) 1))