A166274 -id:A166274 - cp's OEIS frontend

A161924 Permutation of natural numbers: sequence A126441 without zeros.

1, 2, 3, 4, 5, 7, 8, 9, 6, 11, 15, 16, 17, 10, 19, 13, 23, 31, 32, 33, 18, 35, 12, 21, 14, 39, 27, 47, 63, 64, 65, 34, 67, 20, 37, 22, 71, 25, 43, 29, 79, 55, 95, 127, 128, 129, 66, 131, 36, 69, 38, 135, 24, 41, 26, 75, 45, 30, 143, 51, 87, 59, 159, 111, 191, 255, 256

Offset: 1

Alford Arnold, Jun 23 2009

Values appear in the order determined by A004760(n+1)and A062383(n).

The graph of this sequence looks very elegant.

			The table begins:
1.2.4..8.16.32.64.128.256.512.1024
..3.5..9.17.33.65.129.257.513.1025
.......6.10.18.34..66.130.258..514
....7.11.19.35.67.131.259.515.1027
............12.20..36..68.132..260
.........13.21.37..69.133.261..517
............14.22..38..70.134..262
......15.23.39.71.135.263.519.1031
...................24..40..72..136
...............25..41..73.137..265
...................26..42..74..138
............27.43..75.139.267..523
.......................28..44...76
...............29..45..77.141..269
...................30..46..78..142
.........31.47.79.143.271.527.1039
...........................48...80
.......................49..81..145
...........................50...82
...................51..83.147..275
This can be viewed as an irregular table, where row r (>= 1) has A000041(r) elements, that is, as 1; 2,3; 4,5,7; 8,9,6,11,15; 16,17,10,19,13,23,31; etc. A125106 illustrates how each number is mapped to a partition.

Inverse: A166276. a(n) = A126441(A166274(n)). See A161919 for the version with each row sorted into ascending order.

A161511(a(n)) = A036042(n).

columns = 9; row[n_] := n - 2^Floor[Log2[n]]; col[0] = 0; col[n_] := If[EvenQ[n], col[n/2] + DigitCount[n/2, 2, 1], col[(n - 1)/2] + 1]; Clear[T]; T[, ] = 0; Do[T[row[k], col[k]] = k, {k, 1, 2^columns}]; Table[DeleteCases[Table[T[n - 1, k], {n, 1, 2^(k - 1)}], 0], {k, 1, columns}] // Flatten (* Jean-François Alcover, Sep 09 2017 *)

Edited and extended by Antti Karttunen, Oct 12 2009

A126441 Tabular arrangement of the natural numbers: the row on which any nonzero term a(n) appears in is A053645(a(n))=A053645(n+1), and the column is A161511(a(n)). Table is presented by columns with 2^{k-1} items in column k, unused positions are filled with 0's.

Original entry on oeis.org

1, 2, 3, 4, 5, 0, 7, 8, 9, 6, 11, 0, 0, 0, 15, 16, 17, 10, 19, 0, 13, 0, 23, 0, 0, 0, 0, 0, 0, 0, 31, 32, 33, 18, 35, 12, 21, 14, 39, 0, 0, 0, 27, 0, 0, 0, 47, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, 64, 65, 34, 67, 20, 37, 22, 71, 0, 25, 0, 43, 0, 29, 0, 79, 0, 0, 0, 0, 0, 0, 0, 55, 0, 0

Offset: 0

Alford Arnold, Jan 19 2007

Note: 1 might be a more natural starting offset for this sequence, although the identities concerning A053645 and A161511 would have to be changed. - Antti Karttunen, Oct 12 2009.
This can be regarded as an arrangement of the partitions, indexed by position in A125106. The partitions in a given row all have the same remaining partition when the largest part is removed; specifically, the partition indexed by the row number in A125106 (with row 0 having the empty partition remaining).
The first value on row n is A004760(n+1). The second value on each row is A004760(n+1) plus A062383(n); subsequent values increase by ever enlarging powers of two. Or equivalently, each subsequent value on the row after the first nonzero value is given by A004754(previous value on the same row).
A055941(r) tells how many terms the row r (>= 0) has been shifted rightward from its "natural position", i.e. with how many zeros that row has been prepended.
The number of (nonzero) entries in column k is A000041(k).

			The largest power of 2 <= 6 is 4, 6 - 4 = 2, so 6 is in row 2. By A125106, 6 corresponds to the partition [2^2], total 4, so 6 goes in column 4. Thus T(2,4) = 6.
The table begins:
1.2.4..8.16.32.64.128.256.512.1024
..3.5..9.17.33.65.129.257.513.1025
.......6.10.18.34..66.130.258..514
....7.11.19.35.67.131.259.515.1027
............12.20..36..68.132..260
.........13.21.37..69.133.261..517
............14.22..38..70.134..262
......15.23.39.71.135.263.519.1031
...................24..40..72..136
...............25..41..73.137..265
...................26..42..74..138
............27.43..75.139.267..523
.......................28..44...76
...............29..45..77.141..269
...................30..46..78..142
.........31.47.79.143.271.527.1039
...........................48...80
.......................49..81..145
...........................50...82
...................51..83.147..275

A. Karttunen, Table of n, a(n) for n = 0..65534 (first 16 columns)

Cf. A125106, A053645, A000041, A004760, A062383, A000079 (column lengths).
A053645(a(A166274(n))) = A053645(1+A166274(n)) for all n>=1.
Positions of zeros: A166275, this sequence without zeros: A161924. A161920(n) gives the position of the first nonzero term on the row n-1.

columns = 7; row[n_] := n-2^Floor[Log2[n]]; col[0] = 0; col[n_] := If[EvenQ[n], col[n/2] + DigitCount[n/2, 2, 1], col[(n-1)/2]+1]; Clear[T]; T[, ] = 0; Do[T[row[k], col[k]] = k, {k, 1, 2^columns}]; Table[T[n-1, k], {k, 1, columns}, {n, 1, 2^(k-1)}] // Flatten (* Jean-François Alcover, Sep 09 2017 *)

Edited by Franklin T. Adams-Watters, Jan 23 2007

Further edited and Scheme-code added by Antti Karttunen, Oct 12 2009

cp's OEIS Frontend

A161924 Permutation of natural numbers: sequence A126441 without zeros.

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A126441 Tabular arrangement of the natural numbers: the row on which any nonzero term a(n) appears in is A053645(a(n))=A053645(n+1), and the column is A161511(a(n)). Table is presented by columns with 2^{k-1} items in column k, unused positions are filled with 0's.

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A166275 Positions of zeros in A126441.

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