A255554 -id:A255554 - cp's OEIS frontend

A255551 Lucky / Unlucky array, shifted version, read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

2, 4, 3, 6, 5, 7, 8, 11, 19, 9, 10, 17, 39, 27, 13, 12, 23, 61, 57, 45, 15, 14, 29, 81, 91, 97, 55, 21, 16, 35, 103, 121, 147, 117, 85, 25, 18, 41, 123, 153, 199, 181, 177, 109, 31, 20, 47, 145, 183, 253, 243, 277, 225, 139, 33, 22, 53, 165, 217, 301, 315, 369, 345, 295, 157, 37, 24, 59, 187, 247, 351, 379, 471, 465, 447, 325, 175, 43

Offset: 2

Antti Karttunen, Feb 26 2015

Note how in comparison to A255545, the even numbers on the first row have been shifted one step left, "pushing" term 1 out of the array proper. This was done to obtain a better alignment with arrays like A083221 and A255127 associated with other sieves, from which one may then induce permutations like A255553 by cross-referencing.

The starting offset of the sequence giving the terms in square array is 2. However, we can tacitly assume that a(1) = 1 when the sequence is used one-dimensionally as a permutation of natural numbers.

			The top left corner of the array:
   2,   4,   6,   8,  10,  12,  14,   16,   18,   20,   22,   24,   26,   28,   30
   3,   5,  11,  17,  23,  29,  35,   41,   47,   53,   59,   65,   71,   77,   83
   7,  19,  39,  61,  81, 103, 123,  145,  165,  187,  207,  229,  249,  271,  291
   9,  27,  57,  91, 121, 153, 183,  217,  247,  279,  309,  343,  373,  405,  435
  13,  45,  97, 147, 199, 253, 301,  351,  403,  453,  507,  555,  609,  661,  709
  15,  55, 117, 181, 243, 315, 379,  441,  505,  571,  633,  697,  759,  825,  889
  21,  85, 177, 277, 369, 471, 567,  663,  757,  853,  949, 1045, 1141, 1239, 1333
  25, 109, 225, 345, 465, 589, 705,  829,  945, 1063, 1185, 1305, 1423, 1549, 1669
  31, 139, 295, 447, 603, 765, 913, 1075, 1227, 1377, 1537, 1689, 1843, 1999, 2155
  33, 157, 325, 493, 667, 835, 999, 1177, 1347, 1513, 1687, 1861, 2029, 2205, 2367
...

Antti Karttunen, Table of n, a(n) for n = 2..5996; the first 109 antidiagonals of the array, flattened

Inverse: A255552.
Variant of array A255545. (See also A255543).
Row 1: A005843 (even numbers).
Column 1: 2 followed by A000959(2..) (Lucky numbers from their second term onward).
Main diagonal: A255550.
Similar arrays: A083221, A255127.
Associated permutations: A255553, A255554.

(define (A255551 n) (if (<= n 1) n (A255551bi (A002260 (- n 1)) (A004736 (- n 1)))))
(define (A255551bi row col) (cond ((= 1 row) (+ col col)) ((= 1 col) (A000959 row)) (else (A255543bi row (- col 1)))))
;; Other code as in A255543.

For row = 1, A(row,col) = 2*col; For row > 1 and col = 1, A(row,col) = A000959(row); otherwise, A(row,col) = A255543(row,col-1).

A255553 Permutation of natural numbers: a(n) = A255551(A252460(n)).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 9, 8, 5, 10, 13, 12, 15, 14, 11, 16, 21, 18, 25, 20, 17, 22, 31, 24, 19, 26, 23, 28, 33, 30, 37, 32, 29, 34, 39, 36, 43, 38, 35, 40, 49, 42, 51, 44, 41, 46, 63, 48, 27, 50, 47, 52, 67, 54, 61, 56, 53, 58, 69, 60, 73, 62, 59, 64, 81, 66, 75, 68, 65, 70, 79, 72, 87, 74, 71, 76, 57, 78, 93, 80, 77, 82, 99, 84, 103, 86, 83, 88, 105, 90

Offset: 1

Antti Karttunen, Feb 26 2015

nonn

a(n) tells which number in array A255551, constructed from Lucky sieve, is at the same position where n is in array A083221, constructed from the sieve of Eratosthenes. As both arrays have A005843 (even numbers) as their topmost row, this permutation fixes all of them.

Antti Karttunen, Table of n, a(n) for n = 1..6840
Index entries for sequences that are permutations of the natural numbers

Inverse: A255554.
Similar or related permutations: A255407, A255408, A249817, A249818, A252460, A255551.
Cf. A000040, A000959, A001248, A083141, A219178, A255550.

(define (A255553 n) (A255551 (A252460 n)))

a(n) = A255551(A252460(n)).
Other identities:
a(2n) = 2n. [Fixes even numbers.]
For all n >= 1, a(A083141(n)) = A255550(n).
For all n >= 2, a(A000040(n)) = A000959(n).
For all n >= 2, a(A001248(n)) = A219178(n).

A257731 Permutation of natural numbers: a(1) = 1, a(prime(n)) = lucky(1+a(n)), a(composite(n)) = unlucky(a(n)), where prime(n) = n-th prime number A000040, composite(n) = n-th composite number A002808 and lucky = A000959, unlucky = A050505.

Original entry on oeis.org

1, 3, 9, 2, 33, 5, 7, 14, 4, 45, 163, 8, 15, 11, 20, 6, 25, 59, 63, 203, 12, 22, 13, 17, 28, 10, 35, 78, 235, 83, 1093, 251, 18, 30, 19, 24, 31, 39, 16, 47, 67, 101, 43, 290, 107, 1283, 87, 309, 26, 41, 27, 34, 21, 42, 53, 23, 61, 88, 115, 128, 321, 57, 354, 137, 1499, 112, 349, 376, 36, 55, 1401, 38, 49, 46, 29, 56, 70, 32, 99, 81

Offset: 1

Antti Karttunen, May 06 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..6185
Index entries for sequences that are permutations of the natural numbers

Inverse: A257732.
Cf. A000040, A000720, A000959, A002808, A010051, A050505, A065855.
Related or similar permutations: A246377, A255421, A257726, A257733.
Cf. also A032600, A255553, A255554.
Differs from A257733 for the first time at n=19, where a(19) = 63, while A257733(19) = 203.

a(1) = 1; for n > 1: if A010051(n) = 1 [i.e., if n is a prime], then a(n) = A000959(1+a(A000720(n))), otherwise a(n) = A050505(a(A065855(n))).
As a composition of other permutations:
a(n) = A257726(A246377(n)).
a(n) = A257733(A255421(n)).

A257732 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = prime(a(n-1)), a(unlucky(n)) = composite(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and prime = A000040, composite = A002808.

Original entry on oeis.org

1, 4, 2, 9, 6, 16, 7, 12, 3, 26, 14, 21, 23, 8, 13, 39, 24, 33, 35, 15, 53, 22, 56, 36, 17, 49, 51, 25, 75, 34, 37, 78, 5, 52, 27, 69, 101, 72, 38, 102, 50, 54, 43, 106, 10, 74, 40, 94, 73, 134, 83, 98, 55, 135, 70, 76, 62, 141, 18, 100, 57, 125, 19, 99, 175, 114, 41, 130, 167, 77, 176, 95, 89, 104, 137, 86, 184, 28, 149, 133, 80, 164, 30

Offset: 1

Antti Karttunen, May 06 2015

In other words, a(1) = 1 and for n > 1, if n is the k-th lucky number larger than 1 [i.e., n = A000959(k+1)] then a(n) = nthprime(a(k)), otherwise, when n is the k-th unlucky number [i.e., n = A050505(k)], then a(n) = nthcomposite(a(k)).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A257731.
Cf. A000040, A000959, A002808, A050505, A109497, A145649.
Related or similar permutations: A246378, A255422, A257725, A257734.
Cf. also A032600, A255553, A255554.

a(1) = 1; for n > 1: if A145649(n) = 1 [i.e., if n is lucky], then a(n) = A000040(a(A109497(n)-1)), otherwise a(n) = A002808(a(n-A109497(n))).
As a composition of other permutations:
a(n) = A246378(A257725(n)).
a(n) = A255422(A257734(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)).

A255552 Inverse permutation to A255551.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 7, 8, 11, 12, 9, 17, 16, 23, 22, 30, 13, 38, 10, 47, 29, 57, 18, 68, 37, 80, 15, 93, 24, 107, 46, 122, 56, 138, 31, 155, 67, 173, 14, 192, 39, 212, 79, 233, 21, 255, 48, 278, 92, 302, 106, 327, 58, 353, 28, 380, 20, 408, 69, 437, 19, 467, 121, 498, 81, 530, 137, 563, 154, 597, 94, 632, 172, 668, 191, 705, 108, 743, 211, 782, 25

Offset: 1

Antti Karttunen, Feb 26 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..221
Index entries for sequences that are permutations of the natural numbers

Inverse: A255551.

Related permutation: A255554.

cp's OEIS Frontend

A255551 Lucky / Unlucky array, shifted version, read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A255553 Permutation of natural numbers: a(n) = A255551(A252460(n)).

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A257731 Permutation of natural numbers: a(1) = 1, a(prime(n)) = lucky(1+a(n)), a(composite(n)) = unlucky(a(n)), where prime(n) = n-th prime number A000040, composite(n) = n-th composite number A002808 and lucky = A000959, unlucky = A050505.

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A257732 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = prime(a(n-1)), a(unlucky(n)) = composite(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and prime = A000040, composite = A002808.

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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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A255552 Inverse permutation to A255551.

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