A254102 -id:A254102 - cp's OEIS frontend

A126760 a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.

0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 5, 3, 2, 1, 6, 1, 7, 2, 3, 4, 8, 1, 9, 5, 1, 3, 10, 2, 11, 1, 4, 6, 12, 1, 13, 7, 5, 2, 14, 3, 15, 4, 2, 8, 16, 1, 17, 9, 6, 5, 18, 1, 19, 3, 7, 10, 20, 2, 21, 11, 3, 1, 22, 4, 23, 6, 8, 12, 24, 1, 25, 13, 9, 7, 26, 5, 27, 2, 1, 14, 28, 3, 29, 15, 10, 4, 30, 2

Offset: 0

N. J. A. Sloane, Feb 19 2007

nonn

For further information see A126759, which provided the original motivation for this sequence.
From Antti Karttunen, Jan 28 2015: (Start)
The odd bisection of the sequence gives A253887, and the even bisection gives the sequence itself.
A254048 gives the sequence obtained when this sequence is restricted to A007494 (numbers congruent to 0 or 2 mod 3).
For all odd numbers k present in square array A135765, a(k) = the column index of k in that array. (End)
A322026 and this sequence (without the initial zero) are ordinal transforms of each other. - Antti Karttunen, Feb 09 2019
Also ordinal transform of A065331 (after the initial 0). - Antti Karttunen, Sep 08 2024

Antti Karttunen, Table of n, a(n) for n = 0..19683

One less than A126759.
Cf. A003586, A003602, A007310, A064989, A249746, A253887, A254048, A273669, A322317, A323881 (Dirichlet inverse), A323882.
Cf. A347233 (Möbius transform) and also A349390, A349393, A349395 for other Dirichlet convolutions.
Ordinal transform of A065331 and of A322026 (after the initial 0).
Related arrays: A135765, A254102.

f[n_] := Block[{a}, a[0] = 0; a[1] = a[2] = a[3] = 1; a[x_] := Which[EvenQ@ x, a[x/2], Mod[x, 3] == 0, a[x/3], Mod[x, 6] == 1, 2 (x - 1)/6 + 1, Mod[x, 6] == 5, 2 (x - 5)/6 + 2]; Table[a@ i, {i, 0, n}]] (* Michael De Vlieger, Feb 03 2015 *)

A126760(n)={n&&n\=3^valuation(n,3)<M. F. Hasler, Jan 19 2016

a(n) = A126759(n)-1. [The original definition.]
From Antti Karttunen, Jan 28 2015: (Start)
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.
Or with the last clause represented in another way:
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n-1) = 2n.
Other identities. For all n >= 1:
a(n) = A253887(A003602(n)).
a(6n-3) = a(4n-2) = a(2n-1) = A253887(n).
(End)
a(n) = A249746(A003602(A064989(n))). - Antti Karttunen, Feb 04 2015
a(n) = A323882(4*n). - Antti Karttunen, Apr 18 2022

Name replaced with an independent recurrence and the old description moved to the Formula section - Antti Karttunen, Jan 28 2015

A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read 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, 3, 2, 4, 8, 5, 6, 11, 23, 14, 7, 17, 32, 68, 41, 9, 20, 50, 95, 203, 122, 10, 26, 59, 149, 284, 608, 365, 12, 29, 77, 176, 446, 851, 1823, 1094, 13, 35, 86, 230, 527, 1337, 2552, 5468, 3281, 15, 38, 104, 257, 689, 1580, 4010, 7655, 16403, 9842, 16, 44, 113, 311, 770, 2066, 4739, 12029, 22964, 49208, 29525, 18, 47

Offset: 1

L. Edson Jeffery & Antti Karttunen, Jan 24 2015

This is transposed dispersion of (3n-1), starting from its complement A032766 as the first row of square array A(row,col). Please see the transposed array A191450 for references and background discussion about dispersions.

For any odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 -> x (A165355) is found in this array at A(row+1,col).

			The top left corner of the array:
   1,   3,   4,   6,   7,   9,  10,  12,   13,   15,   16,   18,   19,   21
   2,   8,  11,  17,  20,  26,  29,  35,   38,   44,   47,   53,   56,   62
   5,  23,  32,  50,  59,  77,  86, 104,  113,  131,  140,  158,  167,  185
  14,  68,  95, 149, 176, 230, 257, 311,  338,  392,  419,  473,  500,  554
  41, 203, 284, 446, 527, 689, 770, 932, 1013, 1175, 1256, 1418, 1499, 1661
...

Inverse: A254052.
Transpose: A191450.
Row 1: A032766.
Cf. A007051, A057198, A199109, A199113 (columns 1-4).
Cf. A254046 (row index of n in this array, see also A253786), A253887 (column index).
Array A135765(n,k) = 2*A(n,k) - 1.
Other related arrays: A254055, A254101, A254102.
Cf. also A000079, A000244, A003961, A007310, A032766, A002260, A004736, A038722, A165355, A254050.
Related permutations: A048673, A254053, A183209, A249745, A254103, A254104.

In A(n,k)-formulas below, n is the row, and k the column index, both starting from 1:
A(n,k) = (3 + ( A000244(n) * (2*A032766(k) - 1) )) / 6. - Antti Karttunen after L. Edson Jeffery's direct formula for A191450, Jan 24 2015
A(n,k) = A048673(A254053(n,k)). [Alternative formula.]
A(n,k) = (1/2) * (1 + A003961((2^(n-1)) * A254050(k))). [The above expands to this.]
A(n,k) = (1/2) * (1 + (A000244(n-1) * A007310(k))). [Which further reduces to this, equivalent to L. Edson Jeffery's original formula above.]
A(1,k) = A032766(k) and for n > 1: A(n,k) = (3 * A254051(n-1,k)) - 1. [The definition of transposed dispersion of (3n-1).]
A(n,k) = (1+A135765(n,k))/2, or when expressed one-dimensionally, a(n) = (1+A135765(n))/2.
A(n+1,k) = A165355(A135765(n,k)).
As a composition of related permutations. All sequences interpreted as one-dimensional:
a(n) = A048673(A254053(n)). [Proved above.]
a(n) = A191450(A038722(n)). [Transpose of array A191450.]

A135765 Distribute the odd numbers in columns based on the occurrence of "3" in each prime factorization; square array A(row, col) = 3^(row-1) * A007310(col), 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

1, 5, 3, 7, 15, 9, 11, 21, 45, 27, 13, 33, 63, 135, 81, 17, 39, 99, 189, 405, 243, 19, 51, 117, 297, 567, 1215, 729, 23, 57, 153, 351, 891, 1701, 3645, 2187, 25, 69, 171, 459, 1053, 2673, 5103, 10935, 6561, 29, 75, 207, 513, 1377, 3159, 8019, 15309, 32805

Offset: 1

Alford Arnold, Nov 28 2007

The Table can be constructed by multiplying sequence A000244 by A007310.
From Antti Karttunen, Jan 26 2015: (Start)
A permutation of odd numbers. Adding one to each term and then dividing by two gives a related table A254051, which for any odd number, located in this array as x = A(row,col), gives the result at A254051(row+1,col) after one combined Collatz step (3x+1)/2 -> x (A165355) has been applied.
Each odd number n occurs here in position A(A007949(n), A126760(n)).
Compare also to A135764.
(End)

			The top left corner of the array:
    1,    5,    7,   11,   13,   17,   19,   23,   25,   29,   31,   35, ...
    3,   15,   21,   33,   39,   51,   57,   69,   75,   87,   93,  105, ...
    9,   45,   63,   99,  117,  153,  171,  207,  225,  261,  279,  315, ...
   27,  135,  189,  297,  351,  459,  513,  621,  675,  783,  837,  945, ...
   81,  405,  567,  891, 1053, 1377, 1539, 1863, 2025, 2349, 2511, 2835, ...
  243, 1215, 1701, 2673, 3159, 4131, 4617, 5589, 6075, 7047, 7533, 8505, ...
etc.
For n = 6, we have [A002260(6), A004736(6)] = [3, 1] (that is 6 corresponds to location 3,1 (row,col) in above table) and A(3,1) = A000244(3-1) * A007310(1) = 3^2 * 1 = 9.
For n = 9, we have [A002260(9), A004736(9)] = [3, 2] (9 corresponds to location 3,2) and A(3,2) = A000244(3-1) * A007310(2) = 3^2 * 5 = 9*5 = 45.
For n = 13, we have [A002260(13), A004736(13)] = [3, 3] (13 corresponds to location 3,3) and A(3,3) = A000244(3-1) * A007310(3) = 3^2 * 7 = 9*7 = 63.
For n = 23, we have [A002260(23), A004736(23)] = [2, 6] (23 corresponds to location 2,6) and A(2,6) = A000244(2-1) * A007310(6) = 3^1 * 17 = 51.

Row 1: A007310.
Column 1: A000244.
Cf. A007949 (row index), A126760 (column index).
Cf. also A000265, A002260, A003961, A004736, A005408, A165355, A249745.
Related arrays: A135764, A254051, A254055, A254101, A254102.

N:= 20:
B:= [seq(op([6*n+1,6*n+5]),n=0..floor((N-1)/2))]:
[seq(seq(3^j*B[i-j],j=0..i-1),i=1..N)]; # Robert Israel, Jan 26 2015

From Antti Karttunen, Jan 26 2015: (Start)
With both row and col starting from 1:
A(row, col) = A000244(row-1) * A007310(col) = 3^(row-1) * A007310(col).
a(n) = (2*A254051(n))-1.
a(n) = A003961(A254053(n)).
Above in array form:
A(row,col) = A003961(A254053(row,col)) = A003961(A135764(row,A249745(col))).
(End)

Name amended and examples edited by Antti Karttunen, Jan 26 2015

A254055 Square array: A(row,col) = A003602(A254051(row,col)), 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

1, 2, 1, 1, 1, 3, 2, 6, 12, 4, 4, 9, 1, 9, 21, 5, 3, 13, 48, 102, 31, 3, 7, 30, 75, 36, 10, 183, 2, 15, 39, 6, 112, 426, 912, 274, 7, 18, 22, 58, 264, 669, 160, 684, 1641, 8, 10, 7, 129, 345, 198, 1003, 3828, 8202, 2461, 1, 6, 57, 156, 193, 517, 2370, 6015, 2871, 3076, 14763, 5, 24, 66, 85, 117, 1155, 3099, 889, 9022, 34446, 73812, 22144

Offset: 1

Antti Karttunen, Jan 27 2015

Starting with an odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 is found in A254051(row+1,col), and after iterated [i.e., we divide all powers of 2 out] Collatz step: x_new <- A139391(x) = A000265(3x+1) the resulting odd number x_new is located A135764(1,A(row+1,col)).

What the resulting odd number will be, is given by A254101(row+1,col).

			The top left corner of the array:
    1,   2,    1,    2,    4,    5,     3,     2,    7,    8,     1, ...
    1,   1,    6,    9,    3,    7,    15,    18,   10,    6,    24, ...
    3,  12,    1,   13,   30,   39,    22,     7,   57,   66,    18, ...
    4,   9,   48,   75,    6,   58,   129,   156,   85,   25,   210, ...
   21, 102,   36,  112,  264,  345,   193,   117,  507,  588,    79, ...
   31,  10,  426,  669,  198,  517,  1155,  1398,  760,  441,  1884, ...
  183, 912,  160, 1003, 2370, 3099,  1732,    66, 4557, 5286,  1413, ...
  274, 684, 3828, 6015,  889, 4648, 10389, 12576, 6835,  496, 16950, ...
etc.

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array

Cf. A003602, A002260, A004736.

Related arrays: A135764, A135765, A254051, A254101, A254102.

A254101 Square array A(row,col) = A000265(A254051(row,col)).

Original entry on oeis.org

1, 3, 1, 1, 1, 5, 3, 11, 23, 7, 7, 17, 1, 17, 41, 9, 5, 25, 95, 203, 61, 5, 13, 59, 149, 71, 19, 365, 3, 29, 77, 11, 223, 851, 1823, 547, 13, 35, 43, 115, 527, 1337, 319, 1367, 3281, 15, 19, 13, 257, 689, 395, 2005, 7655, 16403, 4921, 1, 11, 113, 311, 385, 1033, 4739, 12029, 5741, 6151, 29525

Offset: 1

Antti Karttunen, Jan 28 2015

Starting with an odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 is found in A254051(row+1,col), and after iterated [i.e., we divide all powers of 2 out] Collatz step: x_new <- A139391(x) = A000265(3x+1) the resulting odd number x_new is located at the first row of array A135764 as x_new = A135764(1,A254055(row+1,col)) and it is given here as A(row+1,col) = A000265(A254051(row+1,col)).

That number's column index in array A135765 is then given by A254102(row+1,col).

			The top left corner of the array:
    1,    3,    1,     3,    7,    9,     5,     3,    13,    15,     1, ...
    1,    1,   11,    17,    5,   13,    29,    35,    19,    11,    47, ...
    5,   23,    1,    25,   59,   77,    43,    13,   113,   131,    35, ...
    7,   17,   95,   149,   11,  115,   257,   311,   169,    49,   419, ...
   41,  203,   71,   223,  527,  689,   385,   233,  1013,  1175,   157, ...
   61,   19,  851,  1337,  395, 1033,  2309,  2795,  1519,   881,  3767, ...
  365, 1823,  319,  2005, 4739, 6197,  3463,   131,  9113, 10571,  2825, ...
  547, 1367, 7655, 12029, 1777, 9295, 20777, 25151, 13669,   991, 33899, ...
etc.

Cf. A000265, A002260, A004736, A003961, A139391, A249745.

Related arrays: A135764, A135765, A254051, A254055, A254102.

(define (A254101 n) (A254101bi (A002260 n) (A004736 n)))
(define (A254101bi row col) (+ -1 (* 2 (A254055bi row col))))
;; Alternative definition:
(define (A254101v2 n) (A254101biv2 (A002260 n) (A004736 n)))
(define (A254101biv2 row col) (A003961 (A254055bi row (A249745 col))))

A(row,col) = A000265(A254051(row,col)).
A(row,col) = (2*A254055(row,col))-1.
A(row,col) = A003961(A254055(row, A249745(col))).
A(row+1,col) = A139391(A135765(row,col)).
As compositions of one-dimensional sequences:
a(n) = A000265(A254051(n)).
a(n) = (2*A254055(n))-1.

A254048 a(n) = A126760(A007494(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 4, 1, 3, 2, 6, 1, 2, 3, 8, 1, 5, 1, 10, 2, 1, 4, 12, 1, 7, 5, 14, 3, 4, 2, 16, 1, 9, 6, 18, 1, 3, 7, 20, 2, 11, 3, 22, 4, 6, 8, 24, 1, 13, 9, 26, 5, 2, 1, 28, 3, 15, 10, 30, 2, 8, 11, 32, 1, 17, 4, 34, 6, 5, 12, 36, 1, 19, 13, 38, 7, 10, 5, 40, 2, 21, 14, 42, 3, 1, 15, 44, 4, 23, 2, 46, 8, 12, 16, 48, 1, 25, 17, 50, 9, 7, 6, 52, 5, 27, 18, 54, 1, 14, 19, 56, 3, 29, 7, 58, 10, 4, 20, 60, 2

Offset: 0

Antti Karttunen, Jan 28 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..19683

Cf. A006370, A007494, A126760, A139391, A253887, A254102.

(define (A254048 n) (A126760 (A007494 n)))

a(n) = A126760(A007494(n)).
Other identities:
a(4n) = A126760(n).
a(4n+1) = A126760(3n+1).
a(4n+2) = A126760(2n+1) = A253887(n+1).
a(4n+3) = 2n+2.
For all n >= 1, a(n) = A126760(A139391(n)). [Conjecture. The proof should be easy. Holds at least up to n = 2^25 = 33554432.]

cp's OEIS Frontend

A126760 a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.

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A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n*(2*floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A135765 Distribute the odd numbers in columns based on the occurrence of "3" in each prime factorization; square array A(row, col) = 3^(row-1) * A007310(col), 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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A254055 Square array: A(row,col) = A003602(A254051(row,col)), 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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A254101 Square array A(row,col) = A000265(A254051(row,col)).

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A254048 a(n) = A126760(A007494(n)).

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A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...