A255127 -id:A255127 - cp's OEIS frontend

A302034 A028234 analog for a factorization process based on the Ludic sieve (A255127); Discard all instances of the (smallest) Ludic factor A272565(n) from n.

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 9, 1, 5, 1, 11, 1, 3, 1, 13, 7, 7, 1, 15, 1, 1, 5, 17, 7, 9, 1, 19, 11, 5, 1, 21, 1, 11, 1, 23, 1, 3, 1, 25, 19, 13, 1, 27, 1, 7, 7, 29, 11, 15, 1, 31, 13, 1, 11, 33, 1, 17, 5, 35, 1, 9, 1, 37, 17, 19, 1, 39, 7, 5, 11, 41, 1, 21, 1, 43, 35, 11, 1, 45, 1, 23, 1, 47, 13, 3, 1, 49, 23, 25, 1, 51, 13, 13, 19

Offset: 1

Antti Karttunen, Apr 01 2018

nonn

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 sequence of distinct Ludic numbers (A003309) in ascending order, while applying A302035 to the same terms gives the corresponding "exponents" of these Ludic factors in this nonstandard "Ludic factorization of n", 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 and examples in A302032.

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences generated by sieves

Cf. A003309, A255127, A260738, A260739, A269379, A269380, A302025, A302026, A302032, A302035.
Cf. A302036 (gives the positions of 1's).
Cf. also A028234, A302044.

\\ Assuming A269379 and its inverse A269380 have been precomputed, then the following is reasonably fast:
A302034(n) = if(1==n,n,my(k=0); while((n%2), n = A269380(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A269379(n); k--); (n));

For n > 1, a(n) = A269379^(r)(A000265(A260739(n))), where r = A260738(n)-1 and A269379^(r)(n) stands for applying r times the map x -> A269379(x), starting from x = n.

a(n) = A302025(A028234(A302026(n))).

A276610 Square array A(row,col) = A255127(row+1,col) - A255127(row,col): the first differences of each column of Ludic array, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 5, 2, 9, 10, 2, 13, 20, 12, 4, 17, 28, 24, 24, 2, 21, 38, 36, 44, 18, 4, 25, 46, 48, 66, 30, 28, 6, 29, 56, 58, 90, 46, 54, 44, 2, 33, 64, 68, 114, 60, 84, 84, 22, 4, 37, 74, 82, 136, 74, 104, 122, 40, 38, 8, 41, 82, 92, 152, 86, 136, 156, 54, 60, 48, 4, 45, 92, 102, 174, 106, 162, 194, 76, 94, 116, 40, 2

Offset: 1

Antti Karttunen, Sep 13 2016

Not all rows are monotonic. See A276620 for their first differences.

			The top left 16 x 15 corner of the array:
1,  5,   9,  13,  17,  21,  25,  29,  33,  37,  41,  45,  49,  53,  57,  61
2, 10,  20,  28,  38,  46,  56,  64,  74,  82,  92, 100, 110, 118, 128, 136
2, 12,  24,  36,  48,  58,  68,  82,  92, 102, 114, 126, 138, 148, 158, 172
4, 24,  44,  66,  90, 114, 136, 152, 174, 202, 222, 244, 264, 284, 310, 330
2, 18,  30,  46,  60,  74,  86, 106, 120, 128, 150, 162, 174, 192, 204, 216
4, 28,  54,  84, 104, 136, 162, 180, 210, 238, 260, 288, 318, 346, 366, 396
6, 44,  84, 122, 156, 194, 234, 282, 316, 348, 388, 428, 464, 504, 548, 584
2, 22,  40,  54,  76,  90, 102, 122, 144, 164, 180, 198, 210, 230, 240, 264
4, 38,  60,  94, 120, 150, 190, 210, 240, 270, 302, 330, 364, 390, 430, 456
8, 48, 116, 162, 236, 288, 336, 406, 446, 510, 576, 622, 680, 738, 786, 844
4, 40,  76, 104, 136, 166, 194, 212, 270, 298, 318, 356, 382, 412, 462, 492
2, 24,  38,  52,  62, 108, 124, 148, 150, 182, 198, 222, 242, 260, 272, 300
4, 38,  70, 116, 148, 164, 210, 240, 270, 300, 354, 388, 414, 448, 474, 504
6, 58, 102, 142, 194, 234, 290, 348, 408, 436, 460, 524, 576, 630, 696, 726
8, 60, 134, 204, 256, 322, 390, 446, 498, 578, 642, 684, 774, 828, 870, 948

Transpose: A276609.
Row 1: A016813.
Column 1: A260723 (from the second 1 onward), Column 2: A276606.
Cf. A255127, A269379.
Cf. also arrays A257257, A257513 and A276620 (gives the first differences of each row).

(define (A276610 n) (A276610bi (A002260 n) (A004736 n)))
(define (A276610bi row col) (- (A255127bi (+ 1 row) col) (A255127bi row col))) ;; Code for A255127bi given in A255127.

A(row,col) = A255127(row+1,col) - A255127(row,col).

A(row,col) = A269379(A255127(row,col)) - A255127(row,col).

A302031 An omega (A001221) analog based on the Ludic sieve (A255127): a(1) = 0; for n > 1, a(n) = 1 + a(A302034(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2

Offset: 1

Antti Karttunen, Apr 02 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences generated by sieves

Cf. A001221, A302026, A302037, A302041,
Cf. A302036 (positions of terms < 2).
Differs from similar A302041 for the first time at n=59, where a(59) = 2, while A302041(59) = 1.

\\ Assuming that A269379 and A269380 have been precomputed:
A302034(n) = if(1==n,n,my(k=0); while((n%2), n = A269380(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A269379(n); k--); (n));
A302031(n) = if(1==n,0,1+A302031(A302034(n)));

a(1) = 0; for n > 1, a(n) = 1 + a(A302034(n)).
a(n) = A001221(A302026(n)).
a(n) = A069010(A269388(n)).

A302037 A bigomega (A001222) analog based on the Ludic sieve (A255127): a(1) = 0; for n > 1, a(n) = 1 + a(A302032(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 2, 3, 3, 2, 1, 4, 1, 2, 2, 3, 1, 3, 2, 5, 3, 2, 2, 4, 1, 3, 2, 4, 1, 4, 1, 3, 4, 2, 1, 5, 3, 2, 3, 3, 1, 3, 2, 4, 3, 2, 2, 4, 1, 3, 2, 6, 2, 4, 1, 3, 4, 3, 1, 5, 2, 2, 2, 4, 1, 3, 3, 5, 3, 2, 1, 5, 3, 2, 3, 4, 1, 5, 1, 3, 5, 2, 2, 6, 1, 4, 2, 3, 2, 4, 2, 4, 4

Offset: 1

Antti Karttunen, Apr 01 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences generated by sieves

Cf. A302026, A302032.
Cf. A003309 (gives the positions of terms <= 1), A302038 (gives the positions of 2's).
Cf. A302031 (an omega-analog), A253557.

\\ Assuming that A269379 and A269380 have been precomputed:
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));
A302037(n) = if(1==n,0,1+A302037(A302032(n)));

a(1) = 0; for n > 1, a(n) = 1 + a(A302032(n)).
a(n) = A000120(A269388(n)).
a(n) = A001222(A302026(n)).

A255415 Row 5 of Ludic array A255127.

Original entry on oeis.org

11, 55, 103, 151, 203, 251, 299, 343, 391, 443, 491, 539, 587, 631, 683, 731, 779, 827, 877, 923, 971, 1019, 1067, 1117, 1165, 1211, 1259, 1307, 1357, 1405, 1453, 1499, 1547, 1597, 1645, 1693, 1741, 1787, 1837, 1885, 1933, 1981, 2033, 2077, 2125, 2173, 2221, 2273, 2321, 2365, 2413, 2461, 2513, 2561, 2609, 2653, 2701, 2753, 2801, 2849, 2897, 2941, 2993, 3041

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

First differences are periodic with period length 48, cf. formulas. - M. F. Hasler, Nov 17 2024

Antti Karttunen, Table of n, a(n) for n = 1..10001

Row 5 of A255127. See A255414 for row 4 and A255416 for row 6.

Cf. A084969, A255407.

L255415=[n*337\14*2+7|n<-[0..47]]+digits(54129937554927109457534, 3)*2
apply( A255415(n)=n--\48*2310+L255415[n%48+1], [1..66]) \\ M. F. Hasler, Nov 10 2024

(define (A255415 n) (A255127bi 5 n)) ;; Code for A255127bi given in A255127.

a(n) = A255407(A084969(n)).

a(n) = a(n-48) + 2310 = a((n-1)%48 + 1) + [(n-1)/48]*2310, where % = mod = remainder operator, and [.] = floor. - M. F. Hasler, Nov 10 2024

A260435 Permutation mapping from Lucky sieve to Ludic sieve: a(1) = 1, for n > 1: a(n) = A255127(A260438(n), A260439(n)).

Original entry on oeis.org

1, 2, 3, 4, 9, 6, 5, 8, 7, 10, 15, 12, 11, 14, 13, 16, 21, 18, 19, 20, 17, 22, 27, 24, 23, 26, 31, 28, 33, 30, 25, 32, 29, 34, 39, 36, 37, 38, 35, 40, 45, 42, 41, 44, 55, 46, 51, 48, 43, 50, 47, 52, 57, 54, 73, 56, 59, 58, 63, 60, 49, 62, 53, 64, 69, 66, 61, 68, 67, 70, 75, 72, 71, 74, 77, 76, 81, 78, 83, 80, 65, 82, 87, 84, 101, 86, 89, 88

Offset: 1

Antti Karttunen, Jul 30 2015

nonn

a(n) tells which number in array A255127 (constructed from Ludic sieve) is at the same position where n is in array A255551 (constructed from Lucky sieve). This permutation fixes all even numbers because both arrays have A005843 as their topmost row.

Inverse: A260436.
Cf. A000959, A003309, A255127, A260438, A260439.
Similar or related permutations: A255407, A255552, A255554, A249817, A249818, A260741 (a more recursed variant).

(define (A260435 n) (if (<= n 1) n (A255127bi (A260438 n) (A260439 n)))) ;; Code for A255127bi given in A255127.

Other identities. For all n >= 1:
a(A000959(n+1)) = A003309(n+2). [Maps Lucky numbers to odd Ludic numbers.]
a(2n) = 2n.
As a composition of related permutations:
a(n) = A255127(A255552(n)).
a(n) = A255407(A255554(n)).

A269384 Permutation of natural numbers: a(1) = 1, a(n) = A255127(A001511(n), a(A003602(n))) - 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 15, 18, 13, 20, 11, 10, 17, 26, 27, 34, 29, 44, 35, 30, 25, 38, 39, 48, 21, 32, 19, 12, 33, 50, 51, 64, 53, 80, 67, 58, 57, 86, 87, 108, 69, 104, 59, 54, 49, 74, 75, 94, 77, 116, 95, 84, 41, 62, 63, 78, 37, 56, 23, 16, 65, 98, 99, 124, 101, 152, 127, 112, 105, 158, 159, 198, 133, 200

Offset: 1

Antti Karttunen, Mar 01 2016

Permutation obtained from the Ludic sieve.
This sequence can be represented as a binary tree. For n > 2, each left hand child is obtained by doubling the contents of the parent node and subtracting one, and each right hand child is obtained by applying A269382(n), when the parent node contains n:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........8                   7......../ \........6
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  9       14         15       18         13       20         11       10
17 26   27  34     29  44   35  30     25  38   39  48     21  32   19  12
etc.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A269383.
Cf. A000035, A001511, A003602, A255127, A269382.
Cf. also A269385, A269387 and also A249814, A269374.

a(1) = 1, a(n) = A255127(A001511(n), a(A003602(n))) - 1.
a(1) = 1, a(2n) = A269382(a(n)), a(2n+1) = (2*a(n+1))-1.
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A276580 Square array A(n,k) = A276570(A255127(n,k)), numbers in Ludic array reduced by the first element of each row. Array is read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 3, 0, 0, 0, 4, 3, 0, 0, 0, 0, 0, 1, 4, 8, 0, 0, 0, 4, 1, 8, 3, 16, 0, 0, 0, 0, 4, 5, 2, 0, 7, 0, 0, 0, 4, 2, 9, 3, 9, 18, 17, 0, 0, 0, 0, 2, 2, 0, 10, 12, 11, 2, 0, 0, 0, 4, 0, 2, 8, 2, 17, 7, 23, 31, 0, 0, 0, 0, 3, 6, 7, 3, 11, 24, 0, 6, 6, 0, 0, 0, 4, 3, 3, 4, 0, 22, 20, 23, 10, 30, 16, 0

Offset: 2

Antti Karttunen, Sep 13 2016

The starting offset of the sequence giving the terms of square array is 2, to tally with the indexing used in A255127. The row and column indices both start from 1.

Row 4 seems to have a period of 8: [0, 3, 3, 1, 1, 4, 2, 2], while row 5 (A276577) seems to have a period of 48.

			The top left 17 x 15 corner of the array:
   n   A003309(n+1) = A255127(n,1).
   |   |
   |   | |           A255127(n,k) modulo A003309(n+1)
   v   v |
   1   2 |0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   2   3 |0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   3   5 |0,  4,  0,  4,  0,  4,  0,  4,  0,  4,  0,  4,  0,  4,  0,  4,  0
   4   7 |0,  3,  3,  1,  1,  4,  2,  2,  0,  3,  3,  1,  1,  4,  2,  2,  0
   5  11 |0,  0,  4,  8,  5,  9,  2,  2,  6,  3,  7,  0,  4,  4,  1,  5,  9
   6  13 |0,  8,  3,  2,  3,  0,  8,  7,  4, 12,  4, 12,  7,  4,  3, 11, 12
   7  17 |0, 16,  0,  9, 10,  2,  3,  0,  7, 10,  0,  3,  8, 13, 12,  0,  5
   8  23 |0,  7, 18, 12, 17, 11, 22, 14,  2,  7,  1, 14,  2, 17,  7, 18, 10
   9  25 |0, 17, 11,  7, 24, 20,  8,  8,  6, 21, 19, 15,  3,  3, 16, 16, 14
  10  29 |0,  2, 23,  0, 23, 25,  0, 25,  0, 25,  2,  2,  0,  2,  6,  8,  8
  11  37 |0, 31,  6, 10, 30, 36,  3, 21, 17, 29, 16, 14, 22, 34,  1, 13, 27
  12  41 |0,  6, 30, 38, 25, 37,  4, 16,  5, 21,  0, 12, 22, 40, 29,  6, 24
  13  43 |0, 16, 42,  9, 35, 38,  7, 31,  8,  1, 25,  6, 24,  5, 35, 26,  1
  14  47 |0, 26, 13, 45, 32, 23, 10, 40, 19, 14, 21,  8, 28, 15,  0, 40, 25
  15  53 |0, 42, 31, 14, 11,  0,  1,  0, 50, 31, 20, 29, 12, 11, 20,  1, 51

Cf. A003309, A255127, A272565, A276570.
Column 1, Rows 1 & 2: A000004.
Column 2: A276576.
Row 5: A276577.

(define (A276580 n) (A276580bi (A002260 (- n 1)) (A004736 (- n 1))))
(define (A276580bi row col) (A276570 (A255127bi row col))) ;; Code for A255127bi given in A255127.

A255128 Inverse permutation to A255127.

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 11, 8, 6, 12, 16, 17, 22, 23, 9, 30, 29, 38, 10, 47, 13, 57, 37, 68, 46, 80, 18, 93, 56, 107, 15, 122, 24, 138, 14, 155, 67, 173, 31, 192, 79, 212, 92, 233, 39, 255, 106, 278, 19, 302, 48, 327, 121, 353, 21, 380, 58, 408, 20, 437, 137, 467, 69, 498, 25, 530, 154, 563, 81, 597, 172, 632, 28, 668, 94, 705, 191, 743, 32

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse: A255127 (with assumed silent a(1)=1).

Cf. A252460, A255408, A255130.

a(n) = A252460(A255408(n)).

A255410 Main diagonal of Ludic array A255127 (and A255129): a(n) = A255127(n,n).

Original entry on oeis.org

2, 9, 35, 85, 203, 325, 547, 911, 1181, 1591, 2347, 2923, 3421, 4151, 5161, 6461, 7693, 8785, 10237, 11789, 13469, 14621, 16523, 19225, 21775, 23669, 25237, 27715, 29891, 34073, 36977, 40487, 43151, 48091, 50429, 53407, 55843, 61541, 68797, 71603, 77279, 80291, 84091, 88771, 91997, 96119, 101927, 108833, 115031, 123187

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024

Cf. A083141, A255127, A255129, A255407.

(define (A255410 n) (A255127bi n n)) ;; Code for A255127bi given in A255127.

a(n) = A255127(n,n).

a(n) = A255407(A083141(n)).

cp's OEIS Frontend

A302034 A028234 analog for a factorization process based on the Ludic sieve (A255127); Discard all instances of the (smallest) Ludic factor A272565(n) from n.

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A276610 Square array A(row,col) = A255127(row+1,col) - A255127(row,col): the first differences of each column of Ludic array, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A302031 An omega (A001221) analog based on the Ludic sieve (A255127): a(1) = 0; for n > 1, a(n) = 1 + a(A302034(n)).

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A302037 A bigomega (A001222) analog based on the Ludic sieve (A255127): a(1) = 0; for n > 1, a(n) = 1 + a(A302032(n)).

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A255415 Row 5 of Ludic array A255127.

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A260435 Permutation mapping from Lucky sieve to Ludic sieve: a(1) = 1, for n > 1: a(n) = A255127(A260438(n), A260439(n)).

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A269384 Permutation of natural numbers: a(1) = 1, a(n) = A255127(A001511(n), a(A003602(n))) - 1.

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A276580 Square array A(n,k) = A276570(A255127(n,k)), numbers in Ludic array reduced by the first element of each row. 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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A255128 Inverse permutation to A255127.