A193231 -id:A193231 - cp's OEIS frontend

A245443 Permutation of nonnegative integers: a(n) = A165199(A193231(n)).

0, 1, 2, 3, 7, 6, 4, 5, 10, 11, 9, 8, 15, 14, 12, 13, 27, 26, 24, 25, 30, 31, 29, 28, 20, 21, 23, 22, 17, 16, 18, 19, 38, 39, 37, 36, 35, 34, 32, 33, 41, 40, 42, 43, 44, 45, 47, 46, 55, 54, 52, 53, 50, 51, 49, 48, 56, 57, 59, 58, 61, 60, 62, 63, 127, 126, 124, 125, 122, 123, 121, 120, 112

Offset: 0

Antti Karttunen, Jul 22 2014

nonn

Inverse: A245444.

Cf. A165199, A193231, A245445, A245446.

(define (A245443 n) (A165199 (A193231 n)))

a(n) = A165199(A193231(n)).

A245444 Permutation of nonnegative integers: a(n) = A193231(A165199(n)).

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 5, 4, 11, 10, 8, 9, 14, 15, 13, 12, 29, 28, 30, 31, 24, 25, 27, 26, 18, 19, 17, 16, 23, 22, 20, 21, 38, 39, 37, 36, 35, 34, 32, 33, 41, 40, 42, 43, 44, 45, 47, 46, 55, 54, 52, 53, 50, 51, 49, 48, 56, 57, 59, 58, 61, 60, 62, 63, 106, 107, 105, 104, 111, 110, 108, 109, 101, 100

Offset: 0

Antti Karttunen, Jul 22 2014

nonn

Inverse: A245443.

Cf. A165199, A193231, A245445, A245446.

(define (A245444 n) (A193231 (A165199 n)))

a(n) = A193231(A165199(n)).

A234745 Blue-code restricted to reducible polynomials over GF(2): a(n) = A193231(A091242(n)).

Original entry on oeis.org

5, 4, 6, 15, 14, 12, 10, 9, 8, 17, 16, 18, 20, 21, 23, 22, 30, 29, 28, 27, 26, 24, 51, 50, 48, 49, 54, 53, 52, 60, 63, 62, 57, 56, 58, 34, 35, 33, 32, 39, 38, 36, 45, 44, 46, 40, 43, 42, 85, 84, 86, 80, 81, 83, 82, 90, 89, 88, 95, 94, 92, 93, 68, 69, 71, 70, 65, 64

Offset: 1

Antti Karttunen, Feb 15 2014

nonn

Cf. A193231, A091242, A234746, A234750.

(define (A234745 n) (A193231 (A091242 n)))

a(n) = A193231(A091242(n)).

A234746 Self-inverse permutation of integers induced by the restriction of blue-code to reducible polynomials over GF(2): a(n) = A091246(A193231(A091242(n))).

Original entry on oeis.org

2, 1, 3, 9, 8, 7, 6, 5, 4, 11, 10, 12, 13, 14, 16, 15, 22, 21, 20, 19, 18, 17, 39, 38, 36, 37, 42, 41, 40, 46, 48, 47, 44, 43, 45, 25, 26, 24, 23, 29, 28, 27, 34, 33, 35, 30, 32, 31, 68, 67, 69, 63, 64, 66, 65, 72, 71, 70, 76, 75, 73, 74, 52, 53, 55, 54, 50, 49, 51, 58, 57, 56, 61, 62, 60, 59

Offset: 1

Antti Karttunen, Feb 15 2014

nonn

Fixed points: A234749.

Cf. A193231, A091242, A091246, A234745, A234751.

(define (A234746 n) (A091246 (A234745 n)))

a(n) = A091246(A234745(n)) = A091246(A193231(A091242(n))).

A234750 Blue-code restricted to irreducible polynomials over GF(2): a(n) = A193231(A014580(n)).

Original entry on oeis.org

3, 2, 7, 13, 11, 19, 31, 25, 55, 61, 59, 37, 47, 41, 87, 91, 67, 73, 103, 97, 109, 117, 115, 253, 241, 247, 239, 229, 203, 193, 211, 213, 171, 167, 185, 191, 157, 145, 137, 143, 131, 285, 283, 319, 313, 301, 299, 351, 333, 357, 355, 361, 375, 369, 379, 505, 499, 501

Offset: 1

Antti Karttunen, Feb 12 2014

The polynomials are encoded as the number whose binary representation is given by the coefficients of the polynomial, e.g., 13 = 2^3 + 2^2 + 2^0 = 1101_2 encodes 1*X^3 + 1*X^2 + 0*X^1 + 1*X^0 = X^3 + X^2 + 1. - Peter Munn, Apr 28 2021

			From _Peter Munn_, Apr 23 2021: (Start)
Table for polynomials of degree less than 6:
   n   A014580   a(n)   Previous 2 columns
          (n)               in binary
   1       2       3         10       11
   2       3       2         11       10
   3       7       7        111      111
   4      11      13       1011     1101
   5      13      11       1101     1011
   6      19      19      10011    10011
   7      25      31      11001    11111
   8      31      25      11111    11001
   9      37      55     100101   110111
  10      41      61     101001   111101
  11      47      59     101111   111011
  12      55      37     110111   100101
  13      59      47     111011   101111
  14      61      41     111101   101001
(End)

Cf. A234751, A193231, A014580.

(define (A234750 n) (A193231 (A014580 n)))

a(n) = A193231(A014580(n)).

A245445 Permutation of nonnegative integers: a(n) = A056539(A193231(n)).

Original entry on oeis.org

0, 1, 3, 2, 5, 6, 4, 7, 15, 8, 12, 11, 10, 13, 9, 14, 17, 30, 22, 25, 26, 21, 29, 18, 16, 31, 23, 24, 27, 20, 28, 19, 51, 44, 60, 35, 36, 59, 43, 52, 48, 47, 63, 32, 39, 56, 40, 55, 46, 49, 33, 62, 57, 38, 54, 41, 45, 50, 34, 61, 58, 37, 53, 42, 85, 106, 74, 117, 122, 69

Offset: 0

Antti Karttunen, Jul 22 2014

nonn

Inverse: A245446.

Cf. A056539, A193231, A245443, A245444.

(define (A245445 n) (A056539 (A193231 n)))

a(n) = A056539(A193231(n)).

A245446 Permutation of nonnegative integers: a(n) = A193231(A056539(n)).

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 5, 7, 9, 14, 12, 11, 10, 13, 15, 8, 24, 16, 23, 31, 29, 21, 18, 26, 27, 19, 20, 28, 30, 22, 17, 25, 43, 50, 58, 35, 36, 61, 53, 44, 46, 55, 63, 38, 33, 56, 48, 41, 40, 49, 57, 32, 39, 62, 54, 47, 45, 52, 60, 37, 34, 59, 51, 42, 126, 84, 77, 103, 111, 69

Offset: 0

Antti Karttunen, Jul 22 2014

nonn

Inverse: A245445.

Cf. A056539, A193231, A245443, A245444.

(define (A245446 n) (A193231 (A056539 n)))

a(n) = A193231(A056539(n)).

A246203 Permutation of natural numbers: a(n) = A246201(A193231(n)).

Original entry on oeis.org

1, 7, 3, 6, 2, 14, 15, 24, 8, 30, 13, 28, 5, 12, 4, 10, 56, 60, 29, 26, 16, 112, 48, 96, 9, 32, 52, 58, 120, 20, 31, 128, 208, 232, 50, 36, 61, 114, 384, 960, 17, 464, 22, 160, 896, 248, 27, 62, 240, 40, 224, 64, 104, 116, 25, 124, 80, 480, 11, 192, 57, 448, 18, 1536, 98, 456, 21, 928, 200, 512, 832, 3584, 121, 244, 144

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

This permutation has the same cycle structure as A246163 has because this is its A193231-conjugate.
On the other hand, it shares with A246201 the following property:
Because 2 is the only even term in A014580, it implies that, apart from a(2)=7, odd numbers occur in odd positions only (along with many even numbers that also occur in odd positions).
Note that for any value k in A246156, "Odd reducible polynomials over GF(2)": 5, 9, 15, 17, 21, 23, ..., a(k) will be even, and apart from 2, all other even numbers are mapped to some even number, so all those terms reside in infinite cycles, and apart from 5 and 15, all of them reside in separate cycles. The infinite cycle containing 5 and 15 goes as: ..., 14523, 3889, 103, 59, 11, 13, 5, 2, 7, 15, 4, 6, 14, 12, 28, 58, 480, 3728, 3932416, ... and it is only because a(2) = 7, that it can temporarily switch back from even terms to odd terms, until right after a(15) = 4 it is finally doomed to the eternal evenness.
See also comments at A246161 and A246163.

Inverse: A246204.
Related permutations: A193231, A246201, A246161, A246163.
Cf. also A000035, A091225, A246156.

(define (A246203 n) (A246201 (A193231 n)))

a(n) = A246201(A193231(n)).
a(n) = A193231(A246163(A193231(n))).
Other identities:
For all n > 1, A000035(a(n)) = A091225(n). [After 1 maps binary representations of reducible GF(2) polynomials to even numbers and the corresponding representations of irreducible polynomials to odd numbers, in some order].

A246204 Permutation of natural numbers: a(n) = A193231(A246202(n)).

Original entry on oeis.org

1, 5, 3, 15, 13, 4, 2, 9, 25, 16, 59, 14, 11, 6, 7, 21, 41, 63, 211, 30, 67, 43, 299, 8, 55, 20, 47, 12, 19, 10, 31, 26, 109, 92, 451, 36, 285, 178, 1615, 50, 253, 108, 1019, 93, 477, 370, 3487, 23, 87, 35, 157, 27, 97, 95, 487, 17, 61, 28, 103, 18, 37, 48, 241, 52, 203, 249, 587, 101, 803, 401, 4591, 83, 369

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

Inverse: A246203.
Related permutations: A193231, A246202, A246164.
Cf. A000035, A091225.

(define (A246204 n) (A193231 (A246202 n)))

a(n) = A193231(A246202(n)).
a(n) = A193231(A246164(A193231(n))).
Other identities:
For all n > 1, A091225(a(n)) = A000035(n). [After 1, maps even numbers to binary representations of reducible GF(2) polynomials and odd numbers to the corresponding representations of irreducible polynomials, in some order. A246202 has the same property].

A265405 Start with a(1) = 1, then always choose for a(n) the least unused number such that A193231(a(n)a(n-1)) = A193231(a(n)) A193231(a(n-1)), where A193231 is an involution of natural numbers called Blue code.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 16, 7, 17, 8, 15, 32, 12, 10, 18, 20, 19, 256, 9, 14, 34, 48, 40, 50, 33, 60, 257, 11, 97, 258, 13, 101, 209, 65536, 21, 259, 64, 30, 65, 51, 80, 24, 84, 36, 85, 66, 260, 22, 4352, 26, 4368, 28, 4369, 37, 768, 41, 770, 42, 771, 68, 90, 272, 45, 273, 56, 1200, 952, 4096, 23, 4097, 27, 4098, 86, 512, 54

Offset: 1

Antti Karttunen, Dec 29 2015

Does this sequence die after a(144) = 46 ?
No, a(145) = 16777216, but whether the sequence is finished remains open. - Rémy Sigrist, Feb 15 2019
The next unused number of the form 2^2^k is always a valid choice, so this sequence is infinite. - Charlie Neder, Apr 14 2019

Rémy Sigrist, Table of n, a(n) for n = 1..183 (first 144 terms from Antti Karttunen) [More terms needed for b-file.]
Rémy Sigrist, PARI program for A265405

Inverse: A265406.
Cf. A193231.
Cf. A266195, A266351, A266405 (sequences with similar definitions, of which at least the first two are known to be infinite and also bijective).

PARI
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cp's OEIS Frontend

A245443 Permutation of nonnegative integers: a(n) = A165199(A193231(n)).

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A245444 Permutation of nonnegative integers: a(n) = A193231(A165199(n)).

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A234745 Blue-code restricted to reducible polynomials over GF(2): a(n) = A193231(A091242(n)).

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A234746 Self-inverse permutation of integers induced by the restriction of blue-code to reducible polynomials over GF(2): a(n) = A091246(A193231(A091242(n))).

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A234750 Blue-code restricted to irreducible polynomials over GF(2): a(n) = A193231(A014580(n)).

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A245445 Permutation of nonnegative integers: a(n) = A056539(A193231(n)).

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A245446 Permutation of nonnegative integers: a(n) = A193231(A056539(n)).

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A246203 Permutation of natural numbers: a(n) = A246201(A193231(n)).

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A246204 Permutation of natural numbers: a(n) = A193231(A246202(n)).

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A265405 Start with a(1) = 1, then always choose for a(n) the least unused number such that A193231(a(n)a(n-1)) = A193231(a(n)) A193231(a(n-1)), where A193231 is an involution of natural numbers called Blue code.