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

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%I A260717 #12 Aug 02 2015 17:54:33
%S A260717 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,
%T A260717 17,9,15,19,23,23,23,23,23,10,17,23,25,25,25,25,25,25,11,19,25,29,29,
%U A260717 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
%N A260717 Square array: row n gives the numbers remaining before the stage n of Ludic sieve.
%C A260717 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.
%C A260717 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.
%H A260717 Antti Karttunen, <a href="/A260717/b260717.txt">Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array</a>
%H A260717 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%e A260717 The top left corner of the array:
%e A260717    2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  14,  15,  16,  17
%e A260717    3,  5,  7,  9, 11, 13, 15, 17,  19,  21,  23,  25,  27,  29,  31,  33
%e A260717    5,  7, 11, 13, 17, 19, 23, 25,  29,  31,  35,  37,  41,  43,  47,  49
%e A260717    7, 11, 13, 17, 23, 25, 29, 31,  37,  41,  43,  47,  53,  55,  59,  61
%e A260717   11, 13, 17, 23, 25, 29, 37, 41,  43,  47,  53,  55,  61,  67,  71,  73
%e A260717   13, 17, 23, 25, 29, 37, 41, 43,  47,  53,  61,  67,  71,  73,  77,  83
%e A260717   17, 23, 25, 29, 37, 41, 43, 47,  53,  61,  67,  71,  77,  83,  89,  91
%e A260717   23, 25, 29, 37, 41, 43, 47, 53,  61,  67,  71,  77,  83,  89,  91,  97
%e A260717   25, 29, 37, 41, 43, 47, 53, 61,  67,  71,  77,  83,  89,  91,  97, 107
%e A260717   29, 37, 41, 43, 47, 53, 61, 67,  71,  77,  83,  89,  91,  97, 107, 115
%e A260717   37, 41, 43, 47, 53, 61, 67, 71,  77,  83,  89,  91,  97, 107, 115, 119
%e A260717   41, 43, 47, 53, 61, 67, 71, 77,  83,  89,  91,  97, 107, 115, 119, 121
%e A260717   43, 47, 53, 61, 67, 71, 77, 83,  89,  91,  97, 107, 115, 119, 121, 127
%e A260717   47, 53, 61, 67, 71, 77, 83, 89,  91,  97, 107, 115, 119, 121, 127, 131
%e A260717   53, 61, 67, 71, 77, 83, 89, 91,  97, 107, 115, 119, 121, 127, 131, 143
%e A260717   61, 67, 71, 77, 83, 89, 91, 97, 107, 115, 119, 121, 127, 131, 143, 149
%e A260717   etc.
%o A260717 (Scheme)
%o A260717 (define (A260717 n) (A260717bi (A002260 n) (A004736 n)))
%o A260717 (define (A260717bi row col) ((rowfun_n_for_A003309sieve row) col))
%o A260717 (define (add1 n) (1+ n))
%o A260717 ;; Uses definec-macro which can memoize also function-closures:
%o A260717 (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)))))))
%Y A260717 Transpose: A260718.
%Y A260717 Column 1: A003309 (without the initial 1).
%Y A260717 Row 1: A020725, Row 2: A144396, Row 3: A007310 (from its second term onward), Row 4: A260714, Row 5: A260715.
%Y A260717 Cf. A255127 (gives the numbers removed at each stage).
%Y A260717 Cf. also array A258207.
%K A260717 nonn,tabl
%O A260717 1,1
%A A260717 _Antti Karttunen_, Jul 30 2015