A079559 -id:A079559

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)).

+20
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

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

Index entries for sequences that are permutations of the natural numbers

FORMULA

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).

PROG

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

CROSSREFS

Inverse: A279342.

Cf. A000120, A070939, A079559, A256992, A256993, A279345.

Related or similar permutations: A054429, A243071, A279338, A279343, A279347.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 10 2016

STATUS

approved

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)).

+20
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

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

Index entries for sequences that are permutations of the natural numbers

FORMULA

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).

PROG

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

CROSSREFS

Inverse: A279344.

Cf. A000120, A070939, A079559, A256992, A256993, A279346.

Related or similar permutations: A054429, A156552, A279338, A279341, A279347.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 10 2016

STATUS

approved

A256991

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

+20
9

0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 5, 6, 7, 7, 8, 8, 9, 10, 9, 10, 11, 12, 11, 13, 14, 12, 13, 14, 15, 15, 16, 16, 17, 18, 17, 18, 19, 20, 19, 21, 22, 20, 21, 22, 23, 24, 23, 25, 26, 24, 25, 27, 28, 26, 29, 30, 27, 28, 29, 30, 31, 31, 32, 32, 33, 34, 33, 34, 35, 36, 35, 37, 38, 36, 37, 38, 39, 40, 39, 41, 42, 40, 41, 43, 44, 42

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

In other words, if n = A005187(k) for some k >= 1, then a(n) = k-1, otherwise it must be that n = A055938(h) for some h, and then a(n) = h.

In binary trees like A233276 and A233278, a(n) gives the contents at the parent node of node containing n, for any n >= 1.

When iterating a(n), a(a(n)), a(a(a(n))), and so on, A070939(n) = A256478(n) + A256479(n) = A257248(n) + A257249(n) gives the number of steps needed to reach zero, from any starting value n >= 1.

LINKS

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

FORMULA

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

a(n) = A256992(n) - A079559(n) = A213714(n) + A234017(n) - A079559(n).

PROG

(Scheme) (define (A256991 n) (if (not (zero? (A079559 n))) (+ -1 (A213714 n)) (A234017 n)))

CROSSREFS

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

Cf. also A256992 (variant), A256478, A256479, A233276, A233278, A257248, A257249.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 15 2015

STATUS

approved

A279338

a(1) = 1, for n > 1, if A079559(n) = 0, a(n) = 2*a(A256992(n)), otherwise a(n) = A003961(a(A256992(n))).

+20
9

1, 2, 3, 5, 4, 6, 7, 9, 10, 15, 11, 8, 12, 14, 25, 21, 18, 35, 13, 20, 30, 27, 45, 22, 33, 49, 16, 24, 28, 50, 55, 75, 42, 77, 17, 36, 70, 63, 105, 26, 125, 175, 40, 60, 54, 39, 65, 90, 121, 81, 44, 66, 135, 99, 98, 147, 91, 32, 48, 56, 100, 110, 245, 165, 150, 143, 19, 84, 154, 225, 231, 34, 275, 385, 72, 140, 126, 51, 343, 210, 539, 189, 52

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A more recursed variant of A279336.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(1) = 1; for n > 1, if A079559(n) = 0 [when n is a term of A055938], a(n) = 2*a(A256992(n)), otherwise a(n) = A003961(a(A256992(n))).

As a composition of other permutations:

a(n) = A163511(A279341(n)).

a(n) = A005940(1+A279343(n)).

a(n) = A250246(A279348(n)).

PROG

(Scheme)

(definec (A279338 n) (cond ((= 1 n) n) ((zero? (A079559 n)) (* 2 (A279338 (A256992 n)))) (else (A003961 (A279338 (A256992 n))))))

CROSSREFS

Inverse: A279339.

Cf. A003961, A055938, A079559, A256992, A278502.

Related or similar permutations: A005940, A163511, A250246, A279336, A279341, A279343, A279348.

KEYWORD

nonn,look

AUTHOR

Antti Karttunen, Dec 10 2016

STATUS

approved

A256478

a(0) = 0; and for n >= 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).

+20
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

a(n) tells how many nonzero terms of A005187 are encountered when traversing toward the root of binary tree A233276, starting from the node containing n. This count includes both n (in case it is a term of A005187) and 1 (but not 0). See also comments in A256479 and A256991.

The 1's (seem to) occur at positions given by A000325.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..16384

FORMULA

a(0) = 0; and for n >= 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).

a(n) = A000120(A233277(n)). [Binary weight of A233277(n).]

Other identities and observations. For all n >= 1:

a(n) = 1 + A257248(n) = 1 + A080791(A233275(n)).

a(n) = A070939(n) - A256479(n).

a(n) >= A255559(n).

PROG

(Scheme, with memoization-macro definec)

(definec (A256478 n) (if (< n 1) n (+ (A079559 n) (A256478 (if (zero? (A079559 n)) (A234017 n) (+ -1 (A213714 n)))))))

;; Alternative definitions:

(define (A256478 n) (A000120 (A233277 n)))

(define (A256478 n) (if (zero? n) n (+ 1 (A080791 (A233275 n)))))

CROSSREFS

One more than A257248.

Cf. A000120, A000325, A005187, A070939, A079559, A080791, A213714, A234017, A233275, A233276, A233277, A255559, A256479, A256991.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 15 2015

STATUS

approved

A256479

a(1) = 0, and for n > 1, if A079559(n) = 0, then a(n) = 1 + a(A234017(n)), otherwise a(n) = a(A213714(n)-1).

+20
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

a(n) tells how many terms of A055938 are encountered when traversing toward the root of binary tree A233276, starting from the node containing n. This count includes also n in case it itself is a term of A055938. See also comments in A256478 and A256991.

LINKS

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

FORMULA

a(1) = 0, and for n > 1, if A079559(n) = 0, then a(n) = 1 + a(A234017(n)), otherwise a(n) = a(A213714(n)-1).

a(n) = A080791(A233277(n)). [Number of nonleading zeros in the binary representation of A233277(n).]

Other identities. For all n >= 1:

a(n) = A257249(n) - 1 = A000120(A233275(n)) - 1.

a(n) = A070939(n) - A256478(n).

a(A000225(n)) = 0.

PROG

(Scheme, with memoization-macro definec)

(definec (A256479 n) (if (<= n 1) 0 (+ (- 1 (A079559 n)) (A256479 (if (zero? (A079559 n)) (A234017 n) (+ -1 (A213714 n)))))))

;; Alternative definitions:

(define (A256479 n) (A080791 (A233277 n)))

(define (A256479 n) (+ -1 (A000120 (A233275 n))))

CROSSREFS

One less than A257249.

Cf. A000120, A055938, A070939, A079559, A080791, A213714, A234017, A233275, A233276, A233277, A256478, A256991.

Cf. also A000225 (gives the positions zeros).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 15 2015

STATUS

approved

A257248

a(1) = 0; and for n > 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).

+20
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,7

COMMENTS

a(n) tells how many nonzero terms of A005187 are encountered when traversing toward the root of binary tree A233276, starting from the node containing n and before 1 is reached. This count includes both n (in case it is a term of A005187) but excludes the 1 and 0 at the root. See also comments in A257249, A256478 and A256991.

LINKS

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

FORMULA

a(1) = 0; and for n > 1, if A079559(n) = 1, then a(n) = 1 + a(A213714(n)-1), otherwise a(n) = a(A234017(n)).

a(n) = A080791(A233275(n)). [Number of nonleading zeros in the binary representation of A233275(n).]

Other identities. For all n >= 1:

a(n) = A256478(n)-1 = A000120(A233277(n))-1.

a(n) = A070939(n) - A257249(n).

PROG

(Scheme, alternative definitions, the first one utilizing memoizing definec-macro)

(definec (A257248 n) (if (= 1 n) 0 (+ (A079559 n) (A257248 (if (zero? (A079559 n)) (A234017 n) (+ -1 (A213714 n)))))))

(define (A257248 n) (- (A256478 n) 1))

CROSSREFS

One less than A256478.

Cf. A000120, A000325, A005187, A070939, A079559, A080791, A213714, A234017, A233275, A233276, A233277, A255559, A257249, A256991.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 19 2015

STATUS

approved

A257249

a(0) = 1, and for n >= 1, if A079559(n) = 0, then a(n) = 1 + a(A234017(n)), otherwise a(n) = a(A213714(n)-1).

+20
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Because A233275(n) = A003188(n) for n = 1 .. 9, a(n) = A005811(n) for n = 1 .. 9.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..8191

FORMULA

a(0) = 1, and for n >= 1, if A079559(n) = 0, then a(n) = 1 + a(A234017(n)), otherwise a(n) = a(A213714(n)-1).

Other identities. For all n >= 1:

a(n) = A070939(n) - A257248(n).

a(n) = A000120(A233275(n)). [Binary weight of A233275(n).]

a(n) = 1 + A256479(n) = 1 + A080791(A233277(n)).

PROG

(Scheme, alternative definitions, the first one utilizing memoizing definec-macro)

(definec (A257249 n) (if (zero? n) 1 (+ (- 1 (A079559 n)) (A257249 (if (zero? (A079559 n)) (A234017 n) (+ -1 (A213714 n)))))))

(define (A257249 n) (+ 1 (A256479 n)))

CROSSREFS

One more than A256479.

Cf. A000120, A003188, A005811, A055938, A070939, A079559, A080791, A213714, A234017, A233275, A233277, A233278, A257248, A256991.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 19 2015

STATUS

approved

A279336

Permutation of natural numbers: a(1) = 1; for n > 1, if A079559(n) = 0, a(n) = 2*A234016(n), otherwise a(n) = A003961(a(A213714(n))).

+20
4

1, 2, 3, 5, 4, 6, 7, 9, 8, 15, 11, 10, 12, 14, 25, 27, 16, 35, 13, 18, 20, 21, 45, 22, 33, 49, 24, 26, 28, 30, 125, 81, 32, 77, 17, 34, 36, 75, 63, 38, 55, 175, 40, 42, 44, 39, 65, 46, 121, 135, 48, 50, 51, 99, 52, 105, 343, 54, 56, 58, 60, 62, 625, 243, 64, 143, 19, 66, 68, 57, 225, 70, 245, 275, 72, 74, 76, 69, 91, 78, 539, 189, 80

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

For n > 1, a(n) = the number which is in the same position of array A246278 where n is located in array A256997.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(1) = 1, for n > 1, if A079559(n) = 0 [when n is a term of A055938], a(n) = 2*A234016(n), otherwise a(n) = A003961(a(A213714(n))).

Other identities:

For all n >= 2, a(n) = A246278(A256998(n)).

PROG

(Scheme)

(definec (A279336 n) (cond ((= 1 n) n) ((zero? (A079559 n)) (* 2 (A234016 n))) (else (A003961 (A279336 (A213714 n))))))

CROSSREFS

Inverse permutation: A279337.

Cf. A003961, A005187, A079559, A213714, A234016, A246278, A256997, A256998.

Cf. also A278501, A279338 (a variant).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 10 2016

STATUS

approved

A279348

a(1) = 1, for n > 1, if A079559(n) = 0, a(n) = 2*a(A256992(n)), otherwise a(n) = A250469(a(A256992(n))).

+20
3

1, 2, 3, 5, 4, 6, 7, 9, 10, 15, 11, 8, 12, 14, 25, 27, 18, 35, 13, 20, 30, 21, 33, 22, 39, 49, 16, 24, 28, 50, 65, 51, 54, 77, 17, 36, 70, 57, 87, 26, 55, 85, 40, 60, 42, 63, 95, 66, 121, 45, 44, 78, 69, 81, 98, 147, 119, 32, 48, 56, 100, 130, 125, 159, 102, 143, 19, 108, 154, 105, 207, 34, 145, 215, 72, 140, 114, 75, 91, 174, 133, 117, 52

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(1) = 1, for n > 1, if A079559(n) = 0 [when n is a term of A055938], a(n) = 2*a(A256992(n)), otherwise a(n) = A250469(a(A256992(n))).

As a composition of other permutations:

a(n) = A250245(A279338(n)).

a(n) = A252753(A279343(n)).

a(n) = A252755(A279341(n)).