A246201 -id:A246201 - cp's OEIS frontend

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

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].

A246377 Permutation of natural numbers: a(1) = 1, a(p_n) = 2a(n)+1, a(c_n) = 2a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n).

Original entry on oeis.org

1, 3, 7, 2, 15, 6, 5, 14, 4, 30, 31, 12, 13, 10, 28, 8, 11, 60, 29, 62, 24, 26, 9, 20, 56, 16, 22, 120, 61, 58, 63, 124, 48, 52, 18, 40, 25, 112, 32, 44, 27, 240, 21, 122, 116, 126, 57, 248, 96, 104, 36, 80, 17, 50, 224, 64, 88, 54, 23, 480, 121, 42, 244, 232, 252, 114, 59, 496, 192, 208, 125, 72, 49, 160, 34, 100

Offset: 1

Antti Karttunen, Aug 27 2014

nonn

This permutation is otherwise like Katarzyna Matylla's A135141, except that the role of even and odd numbers (or alternatively: primes and composites) has been swapped.
Because 2 is the only even prime, it implies that, apart from a(2)=3, odd numbers occur in odd positions only (along with many even numbers that also occur in odd positions).
This also implies that for each odd composite (A071904) there exists a separate infinite cycle in this permutation, apart from 9 and 15 which are in the same infinite cycle: (..., 23, 9, 4, 2, 3, 7, 5, 15, 28, 120, 82, 46, ...).

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

Inverse: A246378.
Cf. A000720, A010051, A065855, A071904, A246348, A246369, A246370.
Other related or similar permutations: A135141, A054429, A246201, A245703, A246376, A246379, A243071, A246681, A236854.
Differs from A237427 for the first time at n=19, where a(19) = 29, while A237427(19) = 62.

a(1) = 1, and for n > 1, if A010051(n) = 1 [i.e. when n is a prime], a(n)  = 1+(2*a(A000720(n))), otherwise a(n) = 2*a(A065855(n)).
As a composition of related permutations:
a(n) = A054429(A135141(n)).
a(n) = A135141(A236854(n)).
a(n) = A246376(A246379(n)).
a(n) = A246201(A245703(n)).
a(n) = A243071(A246681(n)). [For n >= 1].
Other identities.
For all n > 1 the following holds:
A000035(a(n)) = A010051(n). [Maps primes to odd numbers > 1, and composites to even numbers, in some order. Permutations A246379 & A246681 have the same property].

A246202 Permutation of natural numbers: a(1) = 1, a(2n) = A091242(a(n)), a(2n+1) = A014580(a(n)), where A091242(n) = binary code for n-th reducible polynomial over GF(2) and A014580(n) = binary code for n-th irreducible polynomial over GF(2).

Original entry on oeis.org

1, 4, 2, 8, 11, 5, 3, 14, 31, 17, 47, 9, 13, 6, 7, 21, 61, 42, 185, 24, 87, 62, 319, 15, 37, 20, 59, 10, 19, 12, 25, 29, 109, 78, 425, 54, 283, 222, 1627, 33, 131, 108, 647, 79, 433, 373, 3053, 22, 67, 49, 229, 28, 103, 76, 415, 16, 41, 27, 97, 18, 55, 34, 137, 39, 167, 134, 859, 98, 563, 494, 4225, 70, 375, 331, 2705, 264, 2011, 1832, 19891, 44

Offset: 1

Antti Karttunen, Aug 19 2014

This sequence can be represented as a binary tree. Each left hand child is produced as A091242(n), and each right hand child as A014580(n), when the parent contains n:
                                     |
                  ...................1...................
                 4                                       2
       8......../ \.......11                   5......../ \........3
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  14       31         17       47          9       13          6       7
21  61   42  185    24  87   62  319     15 37   20  59      10 19   12 25
etc.
Because 2 is the only even term in A014580, it implies that, apart from a(3)=2, all other odd positions contain an odd number. There are also odd numbers in the even bisection, precisely all the terms of A246156 in some order, together with all even numbers larger than 2 that are also there. See also comments in A246201.

Inverse: A246201.
Similar or related permutations: A245702, A246162, A246164, A246204, A237126, A003188, A054429, A193231, A260422, A260426.
Cf. A014580, A091242, A091225, A246156.

allocatemem((2^31)+(2^30));
uplim = (2^25) + (2^24);
v014580 = vector(2^24);
v091242 = vector(uplim);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < uplim), if(isA014580(n), i++; v014580[i] = n, j++; v091242[j] = n); n++)
A246202(n) = if(1==n, 1, if(0==(n%2), v091242[A246202(n/2)], v014580[A246202((n-1)/2)]));
for(n=1, 638, write("b246202.txt", n, " ", A246202(n)));
\\ Works with PARI Version 2.7.4. - Antti Karttunen, Jul 25 2015
(Scheme, with memoization-macro definec)
(definec (A246202 n) (cond ((< n 2) n) ((odd? n) (A014580 (A246202 (/ (- n 1) 2)))) (else (A091242 (A246202 (/ n 2))))))

a(1) = 1, a(2n) = A091242(a(n)), a(2n+1) = A014580(a(n)).
As a composition of related permutations:
a(n) = A245702(A054429(n)).
a(n) = A246162(A003188(n)).
a(n) = A193231(A246204(n)).
a(n) = A246164(A193231(n)).
a(n) = A260426(A260422(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. A246204 has the same property].

A246163 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A065621(1+a(n)), a(A091242(n)) = A048724(a(n)), where A065621(n) and A048724(n) give the reversing binary representation of n and -n, respectively, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

Original entry on oeis.org

1, 2, 7, 3, 6, 9, 8, 5, 10, 27, 4, 24, 11, 15, 30, 45, 12, 40, 26, 29, 17, 34, 119, 20, 25, 120, 46, 39, 51, 102, 14, 153, 60, 43, 136, 114, 31, 105, 85, 170, 44, 18, 427, 68, 125, 408, 13, 150, 33, 187, 255, 510, 116, 54, 41, 765, 204, 135, 28, 680, 16, 23, 442, 99, 461, 257, 35, 514, 156, 90, 123, 1799, 118, 340, 393, 36

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where the two complementary pairs to be entangled with each other are A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)) and A065621/A048724, the latter which themselves are permutations of A000069/A001969 (odious and evil numbers), which means that this permutation shares many properties with A246161.
Because 3 is the only evil number in A014580, it implies that, apart from a(3)=7, odious numbers occur in odious positions only (along with many evil numbers that also occur in odious positions).
Furthermore, all terms of A246158 reside in infinite cycles, and apart from 4 and 8, all of them reside in separate cycles. The infinite cycle containing 4 and 8 goes as: ..., 2091, 97, 47, 13, 11, 4, 3, 7, 8, 5, 6, 9, 10, 27, 46, 408, 2535, ... and it is only because a(3) = 7, that it can temporarily switch back from evil terms to odious terms, until right after a(8) = 5 it is finally doomed to the eternal evilness.
Please see also the comments at A246201 and A246161.

Inverse: A246164.
Similar or related permutations: A246205, A193231, A246201, A234026, A245701, A234612, A246161, A246203.
Cf. A010060, A014580, A048724, A065621, A091225, A091226, A091245.

a(1) = 1, and for n > 1, if n is in A014580, a(n) = A065621(1+a(A091226(n))), otherwise a(n) = A048724(a(A091245(n))).
As a composition of related permutations:
a(n) = A193231(A246201(n)).
a(n) = A234026(A245701(n)).
a(n) = A234612(A246161(n)).
a(n) = A193231(A246203(A193231(n))).
Other identities:
For all n > 1, A010060(a(n)) = A091225(n). [Maps binary representations of irreducible GF(2) polynomials (A014580) to odious numbers and the corresponding representations of reducible polynomials (A091242) to evil numbers, in some order].

A246161 Permutation of positive integers: a(1) = 1, a(A014580(n)) = A000069(1+a(n)), a(A091242(n)) = A001969(1+a(n)), where A000069 and A001969 are the odious and evil numbers, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 8, 6, 10, 18, 7, 17, 11, 12, 20, 36, 15, 34, 19, 23, 24, 40, 72, 30, 16, 68, 39, 46, 48, 80, 13, 144, 60, 33, 136, 78, 21, 92, 96, 160, 37, 27, 288, 120, 66, 272, 14, 156, 43, 184, 192, 320, 75, 54, 35, 576, 240, 132, 22, 544, 25, 29, 312, 86, 368, 384, 41

Offset: 1

Antti Karttunen, Aug 17 2014

nonn

This is an instance of entanglement permutation, where the two complementary pairs to be entangled with each other are A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)) and A000069/A001969 (odious and evil numbers).
Because 3 is the only evil number in A014580, it implies that, apart from a(3)=4, odious numbers occur in odious positions only (along with many evil numbers that also occur in odious positions).
Note that the two values n=21 and n=35 given in the Example section both encode polynomials reducible over GF(2) and have an odd number of 1-bits in their binary representation (that is, they are both terms of A246158). As this permutation maps all terms of A091242 to the terms of A001969, and apart from a single exception 3 (which here is in a closed cycle: a(3) = 4, a(4) = 3), no term of A001969 is a member of A014580, so they must be members of A091242, thus successive iterations a(21), a(a(21)), a(a(a(21))), etc. always yield some evil number (A001969), so the cycle can never come back to 21 as it is an odious number, so that cycle must be infinite.
On the other hand, when we iterate with the inverse of this permutation, A246162, starting from 21, we see that its successive pre-images 37, 41, 67, 203, 5079 [e.g., 21 = a(a(a(a(a(5079)))))] are all irreducible and thus also odious.
In each such infinite cycle, there can be at most one term which is both reducible (in A091242) and odious (in A000069), i.e. in A246158, thus 21 and 35 must reside in different infinite cycles.
The sequence of fixed points begin as: 1, 2, 5, 19, 54, 71, 73, 865.
Question: apart from them and transposition (3 4) are there any more instances of finite cycles?

			Consider n=21. In binary it is 10101, encoding for polynomial x^4 + x^2 + 1, which factorizes as (x^2 + x + 1)(x^2 + x + 1) over GF(2), in other words, 21 = A048720(7,7). As such, it occurs as the 14th term in A091242, reducible polynomials over GF(2), coded in binary.
By definition of this permutation, a(21) is thus obtained as A001969(1+a(14)). 14 in turn is 8th term in A091242, thus a(14) = A001969(1+a(8)). In turn, 8 = A091242(4), thus a(8) = A001969(1+a(4)), and 4 = A091242(1).
By working the recursion back towards the toplevel, the result is a(21) = A001969(1+A001969(1+A001969(1+A001969(1+1)))) = 24.
Consider n=35. In binary it is 100011, encoding for polynomial x^5 + x + 1, which factorizes as (x^2 + x + 1)(x^3 + x^2 + 1) over GF(2), in other words, 35 = A048720(7,13). As such, it occurs as the 26th term in A091242, thus a(35) = A001969(1+a(26)), and as 26 = A091242(18) and 18 = A091242(12) and 12 = A091242(7) and 7 = A014580(3) [the polynomial x^2 + x + 1 is irreducible over GF(2)], and 3 = A014580(2) and 2 = A014580(1), we obtain the result as a(35) = A001969(1+A001969(1+A001969(1+A001969(1+A000069(1+A000069(1+A000069(2))))))) = 136.

Inverse: A246162.
Related permutations: A003188, A234612, A245701, A233280, A246163, A246201.
Cf. A000069, A001969, A010060, A014580, A091225, A091226, A091242, A091245, A246158.

a(1) = 1, and for n > 1, if n is in A014580, a(n) = A000069(1+a(A091226(n))), otherwise a(n) = A001969(1+a(A091245(n))).
As a composition of related permutations:
a(n) = A233280(A245701(n)).
a(n) = A003188(A246201(n)).
a(n) = A234612(A246163(n)).
Other identities:
For all n > 1, A010060(a(n)) = A091225(n). [Maps binary representations of irreducible GF(2) polynomials (A014580) to odious numbers and the corresponding representations of reducible polynomials (A091242) to evil numbers, in some order].

A260421 a(1) = 1, a(A206074(n)) = 1 + (2a(n)), a(A205783(1+n)) = 2a(n), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

Original entry on oeis.org

1, 3, 7, 2, 15, 6, 5, 14, 4, 30, 31, 12, 13, 10, 28, 8, 11, 60, 29, 62, 24, 26, 9, 20, 61, 56, 16, 22, 63, 120, 25, 58, 124, 48, 52, 18, 27, 40, 122, 112, 21, 32, 57, 44, 126, 240, 17, 50, 116, 248, 96, 104, 23, 36, 121, 54, 80, 244, 59, 224, 125, 42, 64, 114, 88, 252, 49, 480, 53, 34, 19, 100, 41, 232, 496, 192, 123, 208, 113, 46, 33, 72, 45

Offset: 1

Antti Karttunen, Jul 25 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A260422.
Related permutations: A246201, A246377, A260424, A260426.
Cf. A257000, A255572, A255574.

allocatemem(123456789);
uplim = 2^20;
v255574 = vector(uplim); A255574 = n -> v255574[n];
A255572 = n -> (n - A255574(n) - 1);
isA206074(n) = polisirreducible(Pol(binary(n)));
v255574[1] = 0; i=0; j=0; n=2; while((n < uplim), v255574[n] = v255574[n-1]+isA206074(n); n++);
A260421(n) = if(1==n, 1, if(isA206074(n), 1 + 2*(A260421(A255574(n))), 2*(A260421(A255572(n)))));
for(n=1, 8192, write("b260421.txt", n, " ", A260421(n)));

If A257000(n) = 1 [when n is one of the terms of A206074] then a(n) = 1 + 2*a(A255574(n)), otherwise a(n) = 2*A260421(A255572(n)).
As a composition of related permutations:
a(n) = A246377(A260424(n)).
a(n) = A246201(A260426(n)).

A260425 a(1) = 1, a(A014580(n)) = A206074(a(n)), a(A091242(n)) = A205783(1+a(n)), where A014580(n) [resp. A091242(n)] give binary codes for n-th irreducible [resp. reducible] polynomial over GF(2), while A206074 and A205783 give similar codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 15, 7, 10, 13, 16, 21, 26, 14, 18, 19, 22, 27, 34, 40, 24, 11, 30, 32, 35, 42, 51, 23, 60, 38, 20, 46, 49, 31, 52, 63, 76, 43, 36, 92, 57, 33, 68, 17, 74, 48, 78, 95, 114, 64, 54, 25, 135, 86, 50, 37, 102, 47, 28, 111, 72, 118, 140, 67, 165, 96, 82, 39, 195, 79, 128, 75, 56, 150, 70, 44

Offset: 1

Antti Karttunen, Jul 26 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4244
Index entries for sequences that are permutations of the natural numbers

Inverse: A260426.
Related permutations: A246201, A245704, A260422, A260423.
Cf. A014580, A091225, A091226, A091242, A091245, A205783, A206074.
Differs from A245704 for the first time at n=16, where a(16) = 26, while A245704(16) = 25.

allocatemem(234567890);
vecsize = (2^24)-4;
uplim = 2^25;
v091226 = vector(vecsize); A091226 = n -> v091226[n];
A091245(n) = ((n-A091226(n))-1);
v206074 = vector(vecsize); A206074 = n -> v206074[n];
v205783 = vector(vecsize); A205783 = n -> v205783[n];
isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
isA206074(n) = polisirreducible(Pol(binary(n)));
v091226[1] = 0; v205783[1] = 1; i=0; j=1; n=2; while((n < uplim), if(!(n%65536),print1(n,", ")); if(isA206074(n), i++; v206074[i] = n, j++; v205783[j] = n); if(isA014580(n), v091226[n] = v091226[n-1]+1, v091226[n] = v091226[n-1]); n++); print(n);
A260425(n) = if(1==n, 1, if(isA014580(n), A206074(A260425(A091226(n))), A205783(1+A260425(A091245(n)))));
for(n=1, 4244, write("b260425.txt", n, " ", A260425(n)));

(definec (A260425 n) (cond ((<= n 1) n) ((= 1 (A091225 n)) (A206074 (A260425 (A091226 n)))) (else (A205783 (+ 1 (A260425 (A091245 n)))))))

a(1) = 1; for n > 1, if A091225(n) = 1 [when n is in A014580], then a(n) = A206074(a(A091226(n))), otherwise [when n is in A091242], a(n) = A205783(1+a(A091245(n))).
As a composition of related permutations:
a(n) = A260422(A246201(n)).
a(n) = A260423(A245704(n)).

A246205 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A117968(a(n)), a(A091242(n)) = A117967(1+a(n)), where A117967 and A117968 give positive and negative parts of inverse of balanced ternary enumeration of integers, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

Original entry on oeis.org

1, 2, 7, 5, 3, 11, 23, 15, 4, 12, 22, 33, 6, 52, 17, 13, 35, 43, 25, 16, 137, 45, 53, 36, 58, 155, 29, 47, 462, 154, 66, 135, 37, 152, 426, 30, 8, 156, 1273, 428, 24, 148, 460, 41, 423, 1426, 71, 31, 9, 427, 4283, 1410, 34, 431, 75, 1274, 159, 1423, 21, 3707, 194, 99, 44, 10, 1412, 11115, 64, 3850, 38, 1404, 103, 4281, 26, 412, 3722, 49

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

Inverse: A246206.
Similar or related entanglement permutations: A246163, A245701, A246201, A246207, A246209.
Cf. A014580, A091225, A091226, A091242, A091245, A117967, A117968.

a(1) = 1, and for n > 1, if A091225(n) = 1 [i.e. n is in A014580], a(n) = A117968(a(A091226(n))), otherwise a(n) = A117967(1+a(A091245(n))).
As a composition of related permutations:
a(n) = A246207(A245701(n)).
a(n) = A246209(A246201(n)).

cp's OEIS Frontend

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

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A246377 Permutation of natural numbers: a(1) = 1, a(p_n) = 2*a(n)+1, a(c_n) = 2*a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n).

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A246202 Permutation of natural numbers: a(1) = 1, a(2n) = A091242(a(n)), a(2n+1) = A014580(a(n)), where A091242(n) = binary code for n-th reducible polynomial over GF(2) and A014580(n) = binary code for n-th irreducible polynomial over GF(2).

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A246161 Permutation of positive integers: a(1) = 1, a(A014580(n)) = A000069(1+a(n)), a(A091242(n)) = A001969(1+a(n)), where A000069 and A001969 are the odious and evil numbers, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

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A260421 a(1) = 1, a(A206074(n)) = 1 + (2*a(n)), a(A205783(1+n)) = 2*a(n), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

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A246377 Permutation of natural numbers: a(1) = 1, a(p_n) = 2a(n)+1, a(c_n) = 2a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n).

A260421 a(1) = 1, a(A206074(n)) = 1 + (2a(n)), a(A205783(1+n)) = 2a(n), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.