A256993 -id:A256993 - cp's OEIS frontend

A255559 One-based column index of n in array A255555.

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

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

Equally: One-based row index of n in array A255557.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A005187, A055938, A070939, A213714, A255555, A255557.
Cf. also A255560 (corresponding row index).
Cf. A254111, A256989, A256478, A256993.

a(1) = 1; for n > 1, if A213714(n) = 0 [i.e., if n is one of the terms of A055938], then a(n) = 1, otherwise 1 + a(A213714(n)-1).
In other words, a(1) = 1, and for n > 1, if n = A005187(k) for some k, then a(n) = 1 + a(k-1), otherwise it must be that n is in A055938, in which case a(n) = 1.
Other identities and observations. For all n >= 1:
a(n) <= A256478(n) <= A070939(n).
a(n) <= A256993(n) + 1.

A256992 Position of n in either of the complementary sequences, A005187 or A055938: a(n) = A213714(n) + A234017(n).

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 4, 5, 4, 6, 7, 5, 6, 7, 8, 9, 8, 10, 11, 9, 10, 12, 13, 11, 14, 15, 12, 13, 14, 15, 16, 17, 16, 18, 19, 17, 18, 20, 21, 19, 22, 23, 20, 21, 22, 24, 25, 23, 26, 27, 24, 25, 28, 29, 26, 30, 31, 27, 28, 29, 30, 31, 32, 33, 32, 34, 35, 33, 34, 36, 37, 35, 38, 39, 36, 37, 38, 40, 41, 39, 42, 43, 40, 41, 44, 45, 42

Offset: 1

Antti Karttunen, Apr 15 2015

nonn

In other words, if n = A005187(k) for some k >= 1, then a(n) = k, otherwise it must be that n = A055938(h) for some h, and then a(n) = h.
Each n occurs exactly twice, first at a(A005187(n)), then at a(A055938(n)). Cf. also A257126.
When iterating a(n), a(a(n)), a(a(a(n))), etc, A256993(n) gives the number of steps to reach one, from any starting value n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005187, A055938, A079559, A213714, A234017.

Cf. also A256991 (variant), A256993, A257126.

With[{nn = 92}, Function[{g, h}, Flatten@ Table[If[MemberQ[g, n], First@ Position[g, n] - 1, First@ Position[h, n]], {n, Min[Length /@ {g, h}]}]] @@ {Table[2 n - DigitCount[2 n, 2, 1], {n, 0, nn}], Complement[Range@ Last@ #, #] &@ Table[IntegerExponent[(2 n)!, 2], {n, 0, nn}]} ] (* Michael De Vlieger, Dec 12 2016, after Harvey P. Dale at A005187 and Jean-François Alcover at A055938 *)

(define (A256992 n) (+ (A213714 n) (A234017 n)))
(define (A256992 n) (if (not (zero? (A079559 n))) (A213714 n) (A234017 n)))

a(n) = A213714(n) + A234017(n).
a(n) = A256991(n) + A079559(n).
If A079559(n) = 1, a(n) = A213714(n), otherwise a(n) = A234017(n).

A279341 a(1) = 0, a(2) = 1, for n > 2, if A079559(n) = 0, a(n) = 2a(A256992(n)), otherwise a(n) = 1 + 2a(A256992(n)).

Original entry on oeis.org

0, 1, 3, 7, 2, 6, 15, 5, 14, 13, 31, 4, 12, 30, 11, 29, 10, 27, 63, 28, 26, 9, 25, 62, 61, 23, 8, 24, 60, 22, 59, 21, 58, 55, 127, 20, 54, 57, 53, 126, 19, 51, 56, 52, 18, 125, 123, 50, 47, 17, 124, 122, 49, 121, 46, 45, 119, 16, 48, 120, 44, 118, 43, 117, 42, 111, 255, 116, 110, 41, 109, 254, 115, 107, 40, 108, 114, 253, 39, 106, 103

Offset: 1

Antti Karttunen, Dec 10 2016

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

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

Inverse: A279342.
Cf. A000120, A070939, A079559, A256992, A256993, A279345.
Related or similar permutations: A054429, A243071, A279338, A279343, A279347.

(definec (A279341 n) (cond ((<= n 2) (- n 1)) ((zero? (A079559 n)) (* 2 (A279341 (A256992 n)))) (else (+ 1 (* 2 (A279341 (A256992 n)))))))

a(1) = 0, a(2) = 1, for n > 2, if A079559(n) = 0 [when n is a term of A055938], a(n) = 2*a(A256992(n)), otherwise a(n) = 1 + 2*a(A256992(n)).
As a composition of other permutations:
a(n) = A054429(A279343(n)).
a(n) = A279343(A279347(n)).
a(n) = A243071(A279338(n)).
Other identities. For all n >= 1:
A000120(a(n)) = A279345(n).
For all n >= 2, A070939(a(n)) = A256993(n).

A279343 a(1) = 0, and for n > 1, if A079559(n) = 0, a(n) = 1 + 2a(A256992(n)), otherwise a(n) = 2a(A256992(n)).

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 8, 6, 9, 10, 16, 7, 11, 17, 12, 18, 13, 20, 32, 19, 21, 14, 22, 33, 34, 24, 15, 23, 35, 25, 36, 26, 37, 40, 64, 27, 41, 38, 42, 65, 28, 44, 39, 43, 29, 66, 68, 45, 48, 30, 67, 69, 46, 70, 49, 50, 72, 31, 47, 71, 51, 73, 52, 74, 53, 80, 128, 75, 81, 54, 82, 129, 76, 84, 55, 83, 77, 130, 56, 85, 88, 78, 131, 57, 86

Offset: 1

Antti Karttunen, Dec 10 2016

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

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

Inverse: A279344.
Cf. A000120, A070939, A079559, A256992, A256993, A279346.
Related or similar permutations: A054429, A156552, A279338, A279341, A279347.

(definec (A279343 n) (cond ((= 1 n) 0) ((zero? (A079559 n)) (+ 1 (* 2 (A279343 (A256992 n))))) (else (* 2 (A279343 (A256992 n))))))

a(1) = 0, and for n > 1, if A079559(n) = 0 [when n is a term of A055938], a(n) = 1 + 2*a(A256992(n)), otherwise a(n) = 2*a(A256992(n)).
As a composition of other permutations:
a(n) = A054429(A279341(n)).
a(n) = A279341(A279347(n)).
a(n) = A156552(A279338(n)).
Other identities. For all n >= 1:
A000120(a(n)) = A279346(n).
For all n >= 2, A070939(a(n)) = A256993(n).

A256989 One-based column index of n in array A256995.

Original entry on oeis.org

0, 1, 2, 3, 1, 1, 4, 2, 1, 2, 5, 1, 1, 1, 3, 2, 1, 3, 6, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 1, 3, 2, 1, 4, 7, 1, 1, 2, 2, 1, 3, 3, 1, 1, 1, 2, 3, 1, 5, 2, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 1, 1, 3, 2, 1, 5, 8, 1, 1, 2, 2, 1, 3, 3, 1, 1, 1, 2, 4, 1, 4, 2, 1, 1, 2, 2, 1, 3, 4, 1, 1, 1, 1, 2, 6, 1, 3, 2, 1, 1, 2, 3, 1, 3, 2, 1, 1, 1, 3, 5, 1, 2, 2, 1, 1, 2, 2, 1, 2, 4

Offset: 1

Antti Karttunen, Apr 14 2015

nonn

Also one-based row index for array A256997.

a(1) = 0 by convention, as 1 is outside of the actual arrays A256995 & A256997.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A005187, A055938, A213714, A256995, A256997.

Cf. A256990 (corresponding row index), A255559.

a(1) = 0; for n > 1, if A213714(n) = 0 [i.e., if n is one of the terms of A055938], then a(n) = 1, otherwise a(n) = 1 + a(A213714(n)).
In other words, a(1) = 0, and for n > 1, if n = A005187(k) for some k, then a(n) = 1 + a(k), otherwise it must be that n is in A055938, in which case a(n) = 1.
Other observations. For all n >= 1 it holds that:
a(n) <= A256993(n).

A257264 Square array A(row,col) read by antidiagonals: A(1,col) = A055938(col), and for row > 1, A(row,col) = A011371(A(row-1,col)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 03 2015

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
Column n gives the trajectory of iterates of A011371, when starting from A055938(n), thus stepping through successive parent-nodes when starting from the n-th leaf of binary beanstalk, until finally reaching the fixed point 0, which is the root of the whole binary tree.
The hanging tails of columns (upward from the first encountered zero) converge towards A179016.

			The top left corner of the array:
2, 5, 6, 9, 12, 13, 14, 17, 20, 21, 24, 27, 28, 29, 30, 33, 36, 37, 40, 43
1, 3, 4, 7, 10, 10, 11, 15, 18, 18, 22, 23, 25, 25, 26, 31, 34, 34, 38, 39
0, 1, 3, 4,  8,  8,  8, 11, 16, 16, 19, 19, 22, 22, 23, 26, 32, 32, 35, 35
0, 0, 1, 3,  7,  7,  7,  8, 15, 15, 16, 16, 19, 19, 19, 23, 31, 31, 32, 32
0, 0, 0, 1,  4,  4,  4,  7, 11, 11, 15, 15, 16, 16, 16, 19, 26, 26, 31, 31
0, 0, 0, 0,  3,  3,  3,  4,  8,  8, 11, 11, 15, 15, 15, 16, 23, 23, 26, 26
0, 0, 0, 0,  1,  1,  1,  3,  7,  7,  8,  8, 11, 11, 11, 15, 19, 19, 23, 23
0, 0, 0, 0,  0,  0,  0,  1,  4,  4,  7,  7,  8,  8,  8, 11, 16, 16, 19, 19
0, 0, 0, 0,  0,  0,  0,  0,  3,  3,  4,  4,  7,  7,  7,  8, 15, 15, 16, 16
0, 0, 0, 0,  0,  0,  0,  0,  1,  1,  3,  3,  4,  4,  4,  7, 11, 11, 15, 15
0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  1,  1,  3,  3,  3,  4,  8,  8, 11, 11
0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  1,  1,  3,  7,  7,  8,  8
0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  4,  4,  7,  7
0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  3,  3,  4,  4
0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  1,  3,  3
0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  1
...

Row 1: A055938, Row 2: A257507.
Cf. A011371, A055938.
Cf. also A071542, A179016, A218254, A256993, A256997, A257265.

(define (A257264 n) (A257264bi (A002260 n) (A004736 n)))
(define (A257264bi row col) (if (= 1 row) (A055938 col) (A011371 (A257264bi (- row 1) col))))

A279345 a(n) = A000120(A279341(n)).

Original entry on oeis.org

0, 1, 2, 3, 1, 2, 4, 2, 3, 3, 5, 1, 2, 4, 3, 4, 2, 4, 6, 3, 3, 2, 3, 5, 5, 4, 1, 2, 4, 3, 5, 3, 4, 5, 7, 2, 4, 4, 4, 6, 3, 4, 3, 3, 2, 6, 6, 3, 5, 2, 5, 5, 3, 5, 4, 4, 6, 1, 2, 4, 3, 5, 4, 5, 3, 6, 8, 4, 5, 3, 5, 7, 5, 5, 2, 4, 4, 7, 4, 4, 5, 4, 6, 3, 4, 3, 4, 7, 7, 3, 3, 2, 6, 4, 6, 6, 3, 6, 3, 5, 6, 4, 2, 6, 5, 5, 5, 3, 5, 7, 5, 2, 3, 4, 4, 5, 4, 6, 6, 5

Offset: 1

Antti Karttunen, Dec 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000120, A256993, A279341, A279346, A279347.

(define (A279345 n) (A000120 (A279341 n)))

a(n) = A000120(A279341(n)).
a(n) = A279346(A279347(n)).
For all n >= 2, a(n) = 1+A080791(A279343(n)).
For all n >= 2, a(n) + A279346(n) - 1 = A256993(n).

A279346 a(n) = A000120(A279343(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000120, A080791, A279341, A279343, A279345, A279347.

(define (A279346 n) (A000120 (A279343 n)))

a(n) = A000120(A279343(n)).
a(n) = A279345(A279347(n)).
For all n >= 2, a(n) = 1+A080791(A279341(n)).
For all n >= 2, a(n) + A279345(n) - 1 = A256993(n).

cp's OEIS Frontend

A255559 One-based column index of n in array A255555.

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A256992 Position of n in either of the complementary sequences, A005187 or A055938: a(n) = A213714(n) + A234017(n).

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Mathematica

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A279341 a(1) = 0, a(2) = 1, for n > 2, if A079559(n) = 0, a(n) = 2*a(A256992(n)), otherwise a(n) = 1 + 2*a(A256992(n)).

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A279343 a(1) = 0, and for n > 1, if A079559(n) = 0, a(n) = 1 + 2*a(A256992(n)), otherwise a(n) = 2*a(A256992(n)).

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A256989 One-based column index of n in array A256995.

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A257264 Square array A(row,col) read by antidiagonals: A(1,col) = A055938(col), and for row > 1, A(row,col) = A011371(A(row-1,col)).

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A279345 a(n) = A000120(A279341(n)).

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A279346 a(n) = A000120(A279343(n)).

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Formula

A279341 a(1) = 0, a(2) = 1, for n > 2, if A079559(n) = 0, a(n) = 2a(A256992(n)), otherwise a(n) = 1 + 2a(A256992(n)).

A279343 a(1) = 0, and for n > 1, if A079559(n) = 0, a(n) = 1 + 2a(A256992(n)), otherwise a(n) = 2a(A256992(n)).