A260439 -id:A260439

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page 1

A260741

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

+20
11

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

	OFFSET	`1,2`
	COMMENTS	`This is a more recursed variant of A260435.`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..10104` `Index entries for sequences that are permutations of the natural numbers`
	FORMULA	`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.]`
	PROG	`(Scheme, with memoization macro definec)` `(definec (A260741 n) (if (<= n 1) n (A255127bi (A260438 n) (A260741 (A260439 n))))) ;; Code for A255127bi given in A255127.`
	CROSSREFS	`Inverse: A260742.` `Cf. A000959, A003309, A255127, A260438, A260439.` `Similar permutations: A260435, A250245, A250246.`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Jul 30 2015`
	STATUS	`approved`

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)).

+20
9

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

	OFFSET	`1,2`
	COMMENTS	`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.`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..25850`
	FORMULA	`a(1) = 1; for n > 1, a(n) = A255551(A260438(n)+1, A260439(n)).` `Other identities. For all n >= 1:` `A269370(a(n)) = n.`
	PROG	`(Scheme) (define (A269369 n) (if (= 1 n) n (A255551bi (+ (A260438 n) 1) (A260439 n)))) ;; Code for A255551bi given in A255551.`
	CROSSREFS	`Cf. A255551, A260438, A260439, A269372, A269375, A269377.` `Cf. A269370 (left inverse).` `Cf. also A250469, A269379.`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Mar 01 2016`
	STATUS	`approved`

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).

+20
7

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

	OFFSET	`1,3`
	COMMENTS	`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.`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..11017`
	FORMULA	`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.`
	PROG	`(Scheme) (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.`
	CROSSREFS	`Cf. A255551, A260438, A260439, A269369.` `Cf. also A268674, A269380.`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Mar 01 2016`
	STATUS	`approved`

A260435

Permutation mapping from Lucky sieve to Ludic sieve: a(1) = 1, for n > 1: a(n) = A255127(A260438(n), A260439(n)).

+20
5

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

	OFFSET	`1,2`
	COMMENTS	`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.`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..10104` `Index entries for sequences that are permutations of the natural numbers` `Index entries for sequences generated by sieves`
	FORMULA	`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)).`
	PROG	`(Scheme) (define (A260435 n) (if (<= n 1) n (A255127bi (A260438 n) (A260439 n)))) ;; Code for A255127bi given in A255127.`
	CROSSREFS	`Inverse: A260436.` `Cf. A000959, A003309, A255127, A260438, A260439.` `Similar or related permutations: A255407, A255552, A255554, A249817, A249818, A260741 (a more recursed variant).`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Jul 30 2015`
	STATUS	`approved`

A269373

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

+20
4

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

	OFFSET	`1,2`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..1024` `Index entries for sequences that are permutations of the natural numbers`
	FORMULA	`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.]`
	PROG	`(Scheme, with memoization-macro definec)` `(definec (A269373 n) (cond ((<= n 1) n) (else (* (A000079 (- (A260438 (+ 1 n)) 1)) (+ -1 (* 2 (A269373 (A260439 (+ 1 n)))))))))`
	CROSSREFS	`Inverse: A269374.` `Cf. A000035, A000079, A260438, A260439.` `Cf. also A249813, A269383.`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Mar 01 2016`
	STATUS	`approved`

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.

+10
17

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

	OFFSET	`1,4`
	COMMENTS	`Ordinal transform of A272565 (Ludic factor), and also of A260738. - Antti Karttunen, Apr 03 2018`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..10105`
	FORMULA	`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`
	PROG	`(Scheme)` `(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.`
	CROSSREFS	`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.`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Jul 30 2015`
	EXTENSIONS	`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`
	STATUS	`approved`

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.

+10
12

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

	OFFSET	`1,3`
	COMMENTS	`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.`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..10003`
	FORMULA	`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.`
	PROG	`(Scheme)` `(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.`
	CROSSREFS	`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.`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Jul 29 2015`
	STATUS	`approved`

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.

+10
5

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

	OFFSET	`1,2`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..10003`
	FORMULA	`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.`
	PROG	`(Scheme)` `(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.`
	CROSSREFS	`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.`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Jul 29 2015`
	STATUS	`approved`

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