A260439 -id:A260439 - cp's OEIS frontend

A260741 Permutation of natural numbers: a(1) = 1, for n > 1: a(n) = A255127(A260438(n), a(A260439(n))).

1, 2, 3, 4, 9, 6, 5, 8, 7, 18, 15, 12, 11, 10, 13, 16, 21, 14, 19, 36, 17, 30, 51, 24, 23, 22, 31, 20, 33, 26, 25, 32, 29, 42, 27, 28, 37, 38, 35, 72, 45, 34, 41, 60, 55, 102, 39, 48, 43, 46, 47, 44, 105, 62, 73, 40, 59, 66, 87, 52, 49, 50, 53, 64, 69, 58, 61, 84, 67, 54, 63, 56, 71, 74, 77, 76, 57, 70, 83, 144, 125, 90, 75, 68, 101, 82, 89, 120

Offset: 1

Antti Karttunen, Jul 30 2015

nonn

This is a more recursed variant of A260435.

Antti Karttunen, Table of n, a(n) for n = 1..10104
Index entries for sequences that are permutations of the natural numbers

Inverse: A260742.
Cf. A000959, A003309, A255127, A260438, A260439.
Similar permutations: A260435, A250245, A250246.

a(1) = 1, for n > 1: a(n) = A255127(A260438(n), a(A260439(n))).
Other identities. For all n >= 1:
a(A000959(n+1)) = A003309(n+2). [Maps Lucky numbers to odd Ludic numbers.]
a(n) = a(2n)/2. [The even bisection halved gives the sequence back.]

A269369 a(1) = 1, a(n) = A260439(n)-th number k for which A260438(k) = A260438(n)+1; a(n) = A255551(A260438(n)+1, A260439(n)).

Original entry on oeis.org

1, 3, 7, 5, 19, 11, 9, 17, 13, 23, 39, 29, 15, 35, 21, 41, 61, 47, 27, 53, 25, 59, 81, 65, 31, 71, 45, 77, 103, 83, 33, 89, 37, 95, 123, 101, 43, 107, 57, 113, 145, 119, 49, 125, 55, 131, 165, 137, 51, 143, 63, 149, 187, 155, 85, 161, 97, 167, 207, 173, 91, 179, 67, 185, 229, 191, 69, 197, 73, 203, 249, 209, 75

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

For n > 1, a(n) = the number located immediately below n in A255551 (square array generated by Lucky sieve) in the same column where n itself is.

Permutation of odd numbers.

Antti Karttunen, Table of n, a(n) for n = 1..25850

Cf. A255551, A260438, A260439, A269372, A269375, A269377.
Cf. A269370 (left inverse).
Cf. also A250469, A269379.

(define (A269369 n) (if (= 1 n) n (A255551bi (+ (A260438 n) 1) (A260439 n)))) ;; Code for A255551bi given in A255551.

a(1) = 1; for n > 1, a(n) = A255551(A260438(n)+1, A260439(n)).
Other identities. For all n >= 1:
A269370(a(n)) = n.

A269370 a(1) = 1, after which, for odd n: a(n) = A260439(n)-th number k for which A260438(k) = A260438(n)-1, and for even n: a(n) = a(n/2).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 3, 1, 7, 4, 6, 2, 9, 3, 13, 1, 8, 7, 5, 4, 15, 6, 10, 2, 21, 9, 19, 3, 12, 13, 25, 1, 31, 8, 14, 7, 33, 5, 11, 4, 16, 15, 37, 6, 27, 10, 18, 2, 43, 21, 49, 9, 20, 19, 45, 3, 39, 12, 22, 13, 17, 25, 51, 1, 24, 31, 63, 8, 67, 14, 26, 7, 69, 33, 73, 5, 28, 11, 75, 4, 23, 16, 30, 15, 55, 37, 79, 6, 32, 27, 61, 10, 87, 18, 34, 2

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

For odd numbers n > 1, a(n) tells which term is on the immediately preceding row of A255551 (square array generated by Lucky sieve), in the same column where n itself is.

Antti Karttunen, Table of n, a(n) for n = 1..11017

Cf. A255551, A260438, A260439, A269369.

Cf. also A268674, A269380.

(define (A269370 n) (cond ((= 1 n) n) ((even? n) (A269370 (/ n 2))) (else (A255551bi (- (A260438 n) 1) (A260439 n))))) ;; Code for A255551bi given in A255551.

a(1) = 1; after which for even n, a(n) = a(n/2), and for odd n, a(n) = A255551(A260438(n)-1, A260439(n)).
Other identities. For all n >= 1:
a(A269369(n)) = n.

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)).

A269373 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A260438(n+1)-1) * ((2 * a(A260439(n+1))) - 1).

Original entry on oeis.org

1, 2, 3, 6, 5, 4, 11, 8, 9, 10, 7, 16, 21, 32, 15, 22, 17, 12, 19, 64, 13, 18, 31, 128, 41, 24, 63, 14, 29, 256, 43, 512, 33, 42, 23, 1024, 37, 20, 127, 30, 25, 2048, 35, 48, 61, 34, 255, 4096, 81, 8192, 47, 38, 125, 96, 27, 40, 57, 26, 511, 44, 85, 16384, 1023, 62, 65, 32768, 83, 65536, 45, 82, 2047, 131072, 73, 262144, 39

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A269374.
Cf. A000035, A000079, A260438, A260439.
Cf. also A249813, A269383.

a(1) = 1, a(n) = A000079(A260438(n+1)-1) * ((2 * a(A260439(n+1))) - 1).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A260739 Column index to A255127: a(1) = 1; for n > 1, a(n) = the position at the stage where n is removed in the sieve which produces Ludic numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 2, 10, 4, 11, 1, 12, 1, 13, 5, 14, 1, 15, 2, 16, 6, 17, 3, 18, 1, 19, 7, 20, 1, 21, 1, 22, 8, 23, 1, 24, 4, 25, 9, 26, 1, 27, 2, 28, 10, 29, 3, 30, 1, 31, 11, 32, 5, 33, 1, 34, 12, 35, 1, 36, 2, 37, 13, 38, 1, 39, 6, 40, 14, 41, 1, 42, 4, 43, 15, 44, 1, 45, 1, 46, 16, 47, 7, 48, 1, 49, 17, 50, 2

Offset: 1

Antti Karttunen, Jul 30 2015

nonn

Ordinal transform of A272565 (Ludic factor), and also of A260738. - Antti Karttunen, Apr 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..10105

Column index to array A255127.
Cf. A003309, A254100, A272565.
Cf. A260738 (corresponding row index).
Cf. A302035, A302036 (positions of terms that are powers of 2).
Cf. A078898, A246277, A260429, A260439 for column indices to other arrays similar to A255127.
Differs from A246277 (and also after the initial term from A078898) for the first time at n=19.

(define (A260739 n) (cond ((= 1 n) 1) ((even? n) (/ n 2)) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255127bi row col) n) (if (= (A255127bi row col) n) col (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255127bi given in A255127.

Other identities. For all n >= 2:
a(A003309(n)) = 1. [In Ludic sieve each Ludic number (after 1) is the first among the numbers removed at stage k.]
a(A254100(n)) = 2.
A255127(A260738(n), a(n)) = n.
For n > 1, A001511(a(n)) = A302035(n). - Antti Karttunen, Apr 03 2018

Term a(1) changed from 0 to 1 to match with the definition of A078898 and the interpretation as an ordinal transform - Antti Karttunen, Apr 03 2018

A260438 Row index to A255545: If n is k-th Lucky number then a(n) = k, otherwise a(n) = number of the stage where n is removed in Lucky sieve.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 29 2015

nonn

For n >= 2 this works also as a row index to array A255551 (which does not contain 1) and when restricted to unlucky numbers, A050505, also as a row index to array A255543.

Antti Karttunen, Table of n, a(n) for n = 1..10003

Cf. A000959, A050505, A109497, A145649, A219178, A255543, A255545, A255551.
Cf. also A260429, A260439 (corresponding column indices).
Cf. A055396, A260738 for row indices to other arrays similar to A255545.

(define (A260438 n) (cond ((not (zero? (A145649 n))) (A109497 n)) ((even? n) 1) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255543bi row col) n) (if (= (A255543bi row col) n) row (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255543bi given in A255543.

Other identities. For all n >= 1:
a(A000959(n)) = n.
a(A219178(n)) = n.
a(2n) = 1. [All even numbers are removed at the stage one of the sieve.]
a(A016969(n)) = 2.
a(A258016(n)) = 3.
a(A260440(n)) = 4.
A255545(a(n), A260429(n)) = n.
For all n >= 2, A255551(a(n), A260439(n)) = n.

A260429 Column index to A255545: if n is Lucky number, then a(n) = 1, otherwise a(n) = 1 + the position at the stage where n is removed in the Lucky sieve.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 1, 5, 1, 6, 3, 7, 1, 8, 1, 9, 4, 10, 2, 11, 1, 12, 5, 13, 1, 14, 2, 15, 6, 16, 1, 17, 1, 18, 7, 19, 1, 20, 3, 21, 8, 22, 1, 23, 2, 24, 9, 25, 1, 26, 1, 27, 10, 28, 2, 29, 3, 30, 11, 31, 4, 32, 1, 33, 12, 34, 1, 35, 1, 36, 13, 37, 1, 38, 1, 39, 14, 40, 1, 41, 5, 42, 15, 43, 2, 44, 1, 45, 16, 46, 4, 47, 1, 48, 17, 49, 3, 50, 1, 51, 18, 52, 6, 53, 1

Offset: 1

Antti Karttunen, Jul 29 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10003

One more than A260437.
Cf. A000959, A050505, A145649, A219178, A255543, A255545.
Cf. also A260438 (corresponding row index).
Cf. A078898, A246277, A260439, A260739 for column indices to other arrays similar to A255545.

(define (A260429 n) (cond ((not (zero? (A145649 n))) 1) ((even? n) (+ 1 (/ n 2))) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255543bi row col) n) (if (= (A255543bi row col) n) (+ 1 col) (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255543bi given in A255543.

Other identities. For all n >= 1:
a(n) = 1 + A260437(n).
Iff A145649(n) = 1, then a(n) = 1.
a(2n) = n+1. [Even numbers are removed at the stage one of the sieve, after 1 which is also removed in the beginning.]
a(A219178(n)) = 2.
A255545(A260438(n), a(n)) = n.

cp's OEIS Frontend

A260741 Permutation of natural numbers: a(1) = 1, for n > 1: a(n) = A255127(A260438(n), a(A260439(n))).

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A269369 a(1) = 1, a(n) = A260439(n)-th number k for which A260438(k) = A260438(n)+1; a(n) = A255551(A260438(n)+1, A260439(n)).

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A269370 a(1) = 1, after which, for odd n: a(n) = A260439(n)-th number k for which A260438(k) = A260438(n)-1, and for even n: a(n) = a(n/2).

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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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A269373 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A260438(n+1)-1) * ((2 * a(A260439(n+1))) - 1).

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A260739 Column index to A255127: a(1) = 1; for n > 1, a(n) = the position at the stage where n is removed in the sieve which produces Ludic numbers.

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A260438 Row index to A255545: If n is k-th Lucky number then a(n) = k, otherwise a(n) = number of the stage where n is removed in Lucky sieve.

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A260429 Column index to A255545: if n is Lucky number, then a(n) = 1, otherwise a(n) = 1 + the position at the stage where n is removed in the Lucky sieve.

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