A255127 Ludic array: square array A(row,col), where row n lists the numbers removed at stage n in the sieve which produces Ludic numbers. Array is read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...
2, 4, 3, 6, 9, 5, 8, 15, 19, 7, 10, 21, 35, 31, 11, 12, 27, 49, 59, 55, 13, 14, 33, 65, 85, 103, 73, 17, 16, 39, 79, 113, 151, 133, 101, 23, 18, 45, 95, 137, 203, 197, 187, 145, 25, 20, 51, 109, 163, 251, 263, 281, 271, 167, 29, 22, 57, 125, 191, 299, 325, 367, 403, 311, 205, 37, 24, 63, 139, 217, 343, 385, 461, 523, 457, 371, 253, 41
Offset: 2
Examples
The top left corner of the array: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75 5, 19, 35, 49, 65, 79, 95, 109, 125, 139, 155, 169, 185 7, 31, 59, 85, 113, 137, 163, 191, 217, 241, 269, 295, 323 11, 55, 103, 151, 203, 251, 299, 343, 391, 443, 491, 539, 587 13, 73, 133, 197, 263, 325, 385, 449, 511, 571, 641, 701, 761 17, 101, 187, 281, 367, 461, 547, 629, 721, 809, 901, 989, 1079 23, 145, 271, 403, 523, 655, 781, 911, 1037, 1157, 1289, 1417, 1543 25, 167, 311, 457, 599, 745, 883, 1033, 1181, 1321, 1469, 1615, 1753 29, 205, 371, 551, 719, 895, 1073, 1243, 1421, 1591, 1771, 1945, 2117 ...
Links
Crossrefs
Transpose: A255129.
Inverse: A255128. (When considered as a permutation of natural numbers with a(1) = 1).
Main diagonal: A255410.
Row 1: A005843, Row 2: A016945, Row 3: A255413, Row 4: A255414, Row 5: A255415, Row 6: A255416, Row 7: A255417, Row 8: A255418, Row 9: A255419.
A192607 gives all the numbers right of the leftmost column, and A192506 gives the composites among them.
Programs
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Mathematica
rows = 12; cols = 12; t = Range[2, 3000]; r = {1}; n = 1; While[n <= rows, k = First[t]; AppendTo[r, k]; t0 = t; t = Drop[t, {1, -1, k}]; ro[n++] = Complement[t0, t][[1 ;; cols]]]; A = Array[ro, rows]; Table[ A[[n - k + 1, k]], {n, 1, rows}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 14 2016, after Ray Chandler *) -
Python
a255127 = lambda n: A255127(A002260(k-1), A004736(k-1)) def A255127(n, k): A = A255127; R = A.rows while len(R) <= n or len(R[n]) < min(k, A.P[n]): A255127_extend(2*n) return R[n][(k-1) % A.P[n]] + (k-1)//A.P[n] * A.S[n] A=A255127; A.rows=[[1],[2],[3]]; A.P=[1]*3; A.S=[0,2,6]; A.limit=30 def A255127_extend(rMax=9, A=A255127): A.limit *= 2; L = [x+5-x%2 for x in range(0, A.limit, 3)] for r in range(3, rMax): if len(A.P) == r: A.P += [ A.P[-1] * (A.rows[-1][0] - 1) ] # A377469 A.rows += [[]]; A.S += [ A.S[-1] * L[0] ] # ludic factorials if len(R := A.rows[r]) < A.P[r]: # append more terms to this row R += L[ L[0]*len(R) : A.S[r] : L[0] ] L = [x for i, x in enumerate(L) if i%L[0]] # M. F. Hasler, Nov 17 2024
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Scheme
(define (A255127 n) (if (<= n 1) n (A255127bi (A002260 (- n 1)) (A004736 (- n 1))))) (define (A255127bi row col) ((rowfun_n_for_A255127 row) col)) ;; definec-macro memoizes its results: (definec (rowfun_n_for_A255127 n) (if (= 1 n) (lambda (n) (+ n n)) (let* ((rowfun_for_remaining (rowfun_n_for_remaining_numbers (- n 1))) (eka (rowfun_for_remaining 0))) (COMPOSE rowfun_for_remaining (lambda (n) (* eka (- n 1))))))) (definec (rowfun_n_for_remaining_numbers n) (if (= 1 n) (lambda (n) (+ n n 3)) (let* ((rowfun_for_prevrow (rowfun_n_for_remaining_numbers (- n 1))) (off (rowfun_for_prevrow 0))) (COMPOSE rowfun_for_prevrow (lambda (n) (+ 1 n (floor->exact (/ n (- off 1)))))))))
Formula
From M. F. Hasler, Nov 12 2024: (Start)
A(r, c) = A(r, c-P(r)) + S(r) = A(r, ((c-1) mod P(r)) + 1) + floor((c-1)/P(r))*S(r) with periods P = (1, 1, 2, 8, 48, 480, 5760, ...) = A377469, and shifts S = (2, 6, 30, 210, 2310, 30030, 510510) = A376237(2, 3, ...). For example:
A(1, c) = A(1, c-1) + 2 = 2 + (c-1)*2 = 2*c,
A(2, c) = A(2, c-1) + 6 = 3 + (c-1)*6 = 6*c - 3,
A(3, c) = A(3, c-2) + 30 = {5 if c is odd else 19} + floor((c-1)/2)*30 = 15*c - 11 + (c mod 2),
A(4, c) = A(4, c-8) + 210 = A(4, ((c-1) mod 8)+1) + floor((c-1)/8)*210, etc. (End)
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