Search: a260738 -id:a260738
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1, 3, 5, 9, 7, 15, 11, 21, 19, 27, 13, 33, 17, 39, 35, 45, 23, 51, 31, 57, 49, 63, 25, 69, 29, 75, 65, 81, 37, 87, 55, 93, 79, 99, 59, 105, 41, 111, 95, 117, 43, 123, 47, 129, 109, 135, 53, 141, 85, 147, 125, 153, 61, 159, 73, 165, 139, 171, 103, 177, 67, 183, 155, 189, 113, 195, 71, 201, 169, 207, 77, 213, 101, 219, 185, 225, 83
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OFFSET
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1,2
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COMMENTS
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a(n) = the number located immediately below n in A255127 (square array generated by Ludic sieve) in the same column where n itself is, or in other words, the number removed in the next filtering stage at the same step as when n was removed in the A260738(n)-th stage.
Permutation of odd numbers.
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LINKS
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FORMULA
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Other identities. For all n >= 1:
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A269380
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a(1) = 1, after which, for odd numbers: a(n) = A260739(n)-th number k for which A260738(k) = A260738(n)-1, and for even numbers: a(n) = a(n/2).
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+20
14
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1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 9, 3, 8, 7, 17, 2, 23, 11, 10, 5, 25, 6, 19, 1, 12, 13, 15, 4, 29, 9, 14, 3, 37, 8, 41, 7, 16, 17, 43, 2, 21, 23, 18, 11, 47, 10, 31, 5, 20, 25, 35, 6, 53, 19, 22, 1, 27, 12, 61, 13, 24, 15, 67, 4, 55, 29, 26, 9, 71, 14, 33, 3, 28, 37, 77, 8, 49, 41, 30, 7, 83, 16, 89, 17, 32, 43, 39, 2
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OFFSET
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1,3
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COMMENTS
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For odd numbers n > 1, a(n) tells which term is on the immediately preceding row of A255127 (square array generated by Ludic sieve), in the same column where n itself is.
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LINKS
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FORMULA
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a(1) = 1; after which, for even numbers a(n) = a(n/2), and for odd numbers a(n) = A255127(A260738(n)-1, A260739(n)).
Other identities. For all n >= 1:
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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1, 2, 3, 4, 7, 6, 9, 8, 5, 14, 13, 12, 15, 18, 11, 16, 21, 10, 19, 28, 17, 26, 25, 24, 31, 30, 35, 36, 33, 22, 27, 32, 29, 42, 39, 20, 37, 38, 47, 56, 43, 34, 49, 52, 41, 50, 51, 48, 61, 62, 23, 60, 63, 70, 45, 72, 77, 66, 57, 44, 67, 54, 71, 64, 123, 58, 69, 84, 65, 78, 73, 40, 55, 74, 83, 76, 75, 94, 103, 112, 101, 86, 79, 68, 91, 98, 59, 104, 87, 82, 93, 100, 89, 102
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OFFSET
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1,2
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COMMENTS
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This is a more recursed variant of A260436.
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LINKS
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FORMULA
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Other identities. For all n >= 1:
a(A003309(n+2)) = A000959(n+1). [Maps odd Ludic numbers to Lucky numbers.]
a(n) = a(2n)/2. [The even bisection halved gives the sequence back.]
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PROG
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(Scheme, with memoization macro definec)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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1, 2, 3, 4, 7, 6, 9, 8, 5, 10, 13, 12, 15, 14, 11, 16, 21, 18, 19, 20, 17, 22, 25, 24, 31, 26, 23, 28, 33, 30, 27, 32, 29, 34, 39, 36, 37, 38, 35, 40, 43, 42, 49, 44, 41, 46, 51, 48, 61, 50, 47, 52, 63, 54, 45, 56, 53, 58, 57, 60, 67, 62, 59, 64, 81, 66, 69, 68, 65, 70, 73, 72, 55, 74, 71, 76, 75, 78, 103, 80, 77, 82, 79, 84, 91, 86, 83, 88
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OFFSET
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1,2
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COMMENTS
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a(n) tells which number in array A255551 (constructed from Lucky sieve) is at the same position where n is in array A255127 (constructed from Ludic sieve). This permutation fixes all even numbers because both arrays have A005843 as their topmost row.
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LINKS
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FORMULA
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Other identities. For all n >= 1:
a(A003309(n+2)) = A000959(n+1). [Maps odd Ludic numbers to Lucky numbers.]
a(2n) = 2n.
As a composition of related permutations:
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 15, 32, 13, 10, 11, 64, 17, 12, 31, 14, 29, 128, 63, 256, 25, 18, 19, 512, 21, 24, 127, 30, 33, 20, 23, 1024, 61, 26, 27, 2048, 57, 4096, 255, 22, 125, 8192, 511, 28, 49, 34, 35, 16384, 37, 48, 1023, 62, 41, 40, 47, 32768, 253, 58, 59, 36, 65, 65536, 39, 126, 45, 131072, 2047, 96, 121, 50, 51, 262144, 53, 60
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OFFSET
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1,2
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LINKS
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FORMULA
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Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
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PROG
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(Scheme, with memoization-macro definec)
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CROSSREFS
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Differs from both A246683 and A249813 for the first time at n=18, which here a(18)=12, instead of 128.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A255127
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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), ...
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+10
48
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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
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OFFSET
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2,1
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COMMENTS
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The starting offset of the sequence giving the terms of square array is 2. However, we can tacitly assume that a(1) = 1 when the sequence is used as a permutation of natural numbers. However, term 1 itself is out of the array.
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LINKS
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EXAMPLE
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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
...
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MATHEMATICA
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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 *)
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PROG
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(Scheme)
(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)))))))))
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CROSSREFS
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Inverse: A255128. (When considered as a permutation of natural numbers with a(1) = 1).
Cf. A260738 (index of the row where n occurs), A260739 (of the column).
A192607 gives all the numbers right of the leftmost column, and A192506 gives the composites among them.
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KEYWORD
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AUTHOR
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STATUS
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approved
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1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 5, 2, 3, 2, 23, 2, 25, 2, 3, 2, 29, 2, 7, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 5, 2, 3, 2, 53, 2, 11, 2, 3, 2, 7, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 13, 2, 3, 2, 77, 2, 5, 2, 3
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OFFSET
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1,2
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COMMENTS
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This sequence is somewhat analogous to the smallest prime factor of n (A020639). However, each natural number has only one ludic factor, because once it is crossed off in the k-th step of the sieve process, it is not a member of the terms considered in the (k+1)-th step.
On the other hand, by iteratively invoking A302032 it is possible to factor n to its constituent "Ludic factors", with each natural number having a unique such decomposition, analogous to prime factorization of n. See comments and examples given in A302032. - Antti Karttunen, Apr 08 2018
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LINKS
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FORMULA
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(End).
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PROG
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CROSSREFS
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Cf. A276347, A276447, A276448 (ludic factor is equal, less than or greater than the smallest prime factor).
Cf. A260739 (ordinal transform), A302036 (numbers with all Ludic factors equal).
Cf. A264940 (analogous version for lucky numbers).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A260739
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Column index to A255127: a(1) = 1; for n > 1, a(n) = the position at the stage where n is removed in the sieve which produces Ludic numbers.
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+10
17
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1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 2, 10, 4, 11, 1, 12, 1, 13, 5, 14, 1, 15, 2, 16, 6, 17, 3, 18, 1, 19, 7, 20, 1, 21, 1, 22, 8, 23, 1, 24, 4, 25, 9, 26, 1, 27, 2, 28, 10, 29, 3, 30, 1, 31, 11, 32, 5, 33, 1, 34, 12, 35, 1, 36, 2, 37, 13, 38, 1, 39, 6, 40, 14, 41, 1, 42, 4, 43, 15, 44, 1, 45, 1, 46, 16, 47, 7, 48, 1, 49, 17, 50, 2
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OFFSET
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1,4
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COMMENTS
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LINKS
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FORMULA
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Other identities. For all n >= 2:
a(A003309(n)) = 1. [In Ludic sieve each Ludic number (after 1) is the first among the numbers removed at stage k.]
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PROG
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(Scheme)
(define (A260739 n) (cond ((= 1 n) 1) ((even? n) (/ n 2)) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255127bi row col) n) (if (= (A255127bi row col) n) col (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255127bi given in A255127.
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CROSSREFS
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Cf. A260738 (corresponding row index).
Differs from A246277 (and also after the initial term from A078898) for the first time at n=19.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Term a(1) changed from 0 to 1 to match with the definition of A078898 and the interpretation as an ordinal transform - Antti Karttunen, Apr 03 2018
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STATUS
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approved
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A260438
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Row index to A255545: If n is k-th Lucky number then a(n) = k, otherwise a(n) = number of the stage where n is removed in Lucky sieve.
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+10
12
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1, 1, 2, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 3, 1, 7, 1, 2, 1, 8, 1, 4, 1, 2, 1, 9, 1, 10, 1, 2, 1, 11, 1, 3, 1, 2, 1, 12, 1, 5, 1, 2, 1, 13, 1, 14, 1, 2, 1, 6, 1, 4, 1, 2, 1, 3, 1, 15, 1, 2, 1, 16, 1, 17, 1, 2, 1, 18, 1, 19, 1, 2, 1, 20, 1, 3, 1, 2, 1, 7, 1, 21, 1, 2, 1, 4, 1, 22, 1, 2, 1, 5, 1, 23, 1, 2, 1, 3, 1, 24, 1, 2, 1, 8, 1, 25, 1, 2, 1, 26, 1, 6, 1, 2, 1
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OFFSET
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1,3
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COMMENTS
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For n >= 2 this works also as a row index to array A255551 (which does not contain 1) and when restricted to unlucky numbers, A050505, also as a row index to array A255543.
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LINKS
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FORMULA
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Other identities. For all n >= 1:
a(2n) = 1. [All even numbers are removed at the stage one of the sieve.]
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PROG
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(Scheme)
(define (A260438 n) (cond ((not (zero? (A145649 n))) (A109497 n)) ((even? n) 1) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255543bi row col) n) (if (= (A255543bi row col) n) row (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255543bi given in A255543.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A302032
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A032742 analog for a nonstandard factorization process based on the Ludic sieve (A255127); Discard a single instance of the Ludic factor A272565(n) from n.
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+10
11
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1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 5, 10, 9, 11, 1, 12, 1, 13, 7, 14, 1, 15, 7, 16, 15, 17, 7, 18, 1, 19, 11, 20, 1, 21, 1, 22, 21, 23, 1, 24, 19, 25, 19, 26, 1, 27, 11, 28, 27, 29, 11, 30, 1, 31, 13, 32, 11, 33, 1, 34, 33, 35, 1, 36, 13, 37, 17, 38, 1, 39, 35, 40, 39, 41, 1, 42, 31, 43, 35, 44, 1, 45, 1, 46, 45, 47, 13, 48, 1, 49, 23, 50
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OFFSET
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1,4
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COMMENTS
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Like [A020639(n), A032742(n)] or [A020639(n), A302042(n)], also ordered pair [A272565(n), a(n)] is unique for each n. Iterating n, a(n), a(a(n)), a(a(a(n))), ..., until 1 is reached, and taking the Ludic factor (A272565) of each term gives a multiset of Ludic numbers (A003309) in ascending order, unique for each natural number n >= 1. Permutation pair A302025/A302026 maps between this "Ludic factorization" and the ordinary prime factorization of n. See also comments in A302034.
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LINKS
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FORMULA
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EXAMPLE
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For n = 100, A272565(100) [its Ludic factor] is 2. Because A260738(100) = 1, a(100) is just A260739(100) = 100/2 = 50.
For n = 50, A272565(50) [its Ludic factor] is 2. Because A260738(50) = 1, a(50) = A260739(50) = 50/2 = 25.
Collecting the Ludic factors given by A272565 we get a multiset of factors: [2, 2, 25] = [A003309(1+1), A003309(1+1), A003309(1+9)]. Note that prime(1)*prime(1)*prime(9) = 2*2*23 = 92 = A302026(100).
If we start from n = 100, iterating the map n -> A302034(n) [instead of n -> A302032(n)] and apply A272565 to each term obtained we get just a single instance of each Ludic factor: [2, 25]. Then by applying A302035 to the same terms we get the corresponding exponents (multiplicities) of those factors: [2, 1].
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PROG
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(PARI)
\\ Assuming A269379 and its inverse A269380 have been precomputed, then the following is reasonably fast:
A302032(n) = if(1==n, n, my(k=0); while((n%2), n = A269380(n); k++); n = n/2; while(k>0, n = A269379(n); k--); (n));
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CROSSREFS
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Cf. A003309, A255127, A269379, A269380, A272565, A260738, A260739, A269379, A269380, A302025, A302026, A302034, A302035.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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