A091242 -id:A091242 - cp's OEIS frontend

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).

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].

A246162 Permutation of natural numbers: a(1) = 1, a(A000069(n)) = A014580(a(n-1)), a(A001969(n)) = A091242(a(n-1)), 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, 8, 11, 7, 6, 9, 13, 14, 31, 47, 17, 25, 12, 10, 19, 15, 37, 59, 20, 21, 61, 185, 42, 319, 62, 24, 87, 137, 34, 18, 55, 16, 41, 97, 27, 22, 67, 229, 49, 415, 76, 28, 103, 29, 109, 425, 78, 1627, 222, 54, 283, 3053, 373, 79, 433, 33, 131, 647, 108, 1123, 166, 45, 203, 26, 91, 379, 71, 23

Offset: 1

Antti Karttunen, Aug 17 2014. Erroneous comment corrected Aug 20 2014

nonn

This is an instance of entanglement-permutation, where the two complementary pairs to be entangled with each other are A000069/A001969 (odious and evil numbers) and A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)).

Because 3 is the only evil number in A014580, it implies that, apart from a(4)=3, all other odious positions contain an odious number. There are also odious numbers in some of the evil positions, precisely all the terms of A246158 in some order, together with all evil numbers larger than 3. (Permutation A246164 has the same property, except there a(7)=3.) See comments in A246161 for more details how this affects the cycle structure of these permutations.

Inverse: A246161.
Related permutations: A006068, A233279, A234612, A237427, A245702, A246202, A246164.
Cf. A010060, A014580, A091242, A115384, A245710.

a(1) = 1, and for n > 1, if A010060(n) = 1 [i.e. n is one of the odious numbers, A000069], a(n) = A014580(a(A115384(n)-1)), otherwise, a(n) = A091242(a(A245710(n))).
As a composition of related permutations:
a(n) = A245702(A233279(n)).
a(n) = A246202(A006068(n)).
a(n) = A246164(A234612(n)).
For all n > 1, A091225(a(n)) = A010060(n). [Maps odious numbers to binary representations of irreducible GF(2) polynomials (A014580) and evil numbers to the corresponding representations of reducible polynomials (A091242), in some order].

A091246 Inverse function of A091242: position in A091242 if the n-th GF(2)[X] polynomial is reducible, 0 otherwise.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 03 2004

nonn

Analogous to A066246.

Inverse of A091242.

a(n) = A091245(n) * A091247(n).

A245813 Permutation of natural numbers induced when A091205 is restricted to {1} and binary codes for polynomials reducible over GF(2): a(1) = 1, a(n) = A062298(A091205(A091242(n-1))).

Original entry on oeis.org

1, 2, 5, 3, 4, 9, 11, 7, 6, 18, 10, 59, 20, 25, 16, 8, 50, 15, 32, 31, 12, 13, 38, 21, 41, 125, 85, 43, 17, 45, 52, 35, 22, 19, 103, 105, 33, 24, 14, 190, 68, 27, 66, 28, 161, 29, 80, 26, 54, 46, 177, 84, 258, 34, 180, 64, 90, 70, 507, 37, 196, 96, 39, 110, 430, 92, 78, 75, 600, 48, 40, 82, 213, 218, 71, 23, 87, 72, 51, 132, 30

Offset: 1

Antti Karttunen, Aug 16 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245814.
Cf. A062298, A091242.
Related permutations: A091205, A245815, A245820.

allocatemem(234567890);
v091226 = vector(2^22);
v091242 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; v091226[n] = v091226[n-1]+1, j++; v091242[j] = n; v091226[n] = v091226[n-1]); n++);
A062298(n) = n-primepi(n);
A091226(n) = v091226[n];
A091242(n) = v091242[n];
A091205(n) = if(n<=1, n, if(isA014580(n), prime(A091205(A091226(n))), {my(irfs, t); irfs=subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2); irfs[,1]=apply(t->A091205(t), irfs[,1]); factorback(irfs)}));
A245813(n) = if(n<=1, n, A062298(A091205(A091242(n-1))));
for(n=1, 10001, write("b245813.txt", n, " ", A245813(n)));

(define (A245813 n) (if (<= n 1) n (A062298 (A091205 (A091242 (- n 1))))))

a(1) = 1, and for n > 1, a(n) = A062298(A091205(A091242(n-1))).
As a composition of related permutations:
a(n) = A245815(A245820(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))).

A245820 Permutation of natural numbers induced when A245704 is restricted to {1} and binary codes for polynomials reducible over GF(2): a(1) = 1, a(n) = A062298(A245704(A091242(n-1))).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 6, 10, 13, 16, 8, 11, 14, 17, 22, 26, 15, 19, 20, 23, 27, 34, 39, 25, 12, 29, 31, 35, 40, 50, 24, 56, 37, 21, 43, 46, 38, 51, 57, 70, 48, 36, 78, 53, 33, 61, 18, 65, 55, 71, 79, 95, 67, 52, 30, 106, 75, 49, 42, 85, 54, 28, 89, 77, 96, 107, 74, 126, 92, 73, 45, 141, 98, 101, 69, 59, 116, 76, 41, 120, 105

Offset: 1

Antti Karttunen, Aug 16 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245819.
Related permutations: A245704, A245813, A245816.
Cf. A014580, A036234, A062298, A091242.

\\ The rest of code can be found in A245704.
A245820(n) = if(1==n, 1, 1 + A245704(n-1));

(define (A245820 n) (if (<= n 1) n (+ 1 (A245704 (- n 1)))))

(define (A245820 n) (if (<= n 1) n (A062298 (A245704 (A091242 (- n 1))))))

(define (A245820 n) (if (<= n 1) n (A036234 (A245704 (A014580 (- n 1))))))

a(1) = 1, and for n > 1, a(n) = 1 + A245704(n-1).
a(1) = 1, and for n > 1, a(n) = A062298(A245704(A091242(n-1))). [Induced when A245704 is restricted to {1} and binary codes for polynomials reducible over GF(2)].
a(1) = 1, and for n > 1, a(n) = A036234(A245704(A014580(n-1))). [Induced also when A245703 is restricted to {1} and other binary codes for polynomials not reducible over GF(2)].
As a composition of related permutations:
a(n) = A245816(A245813(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)).

A246206 Permutation of natural numbers: a(1) = 1, if A117966(n) < 0, a(n) = A014580(a(-(A117966(n)))), otherwise a(n) = A091242(a(A117966(n)-1)).

Original entry on oeis.org

1, 2, 5, 9, 4, 13, 3, 37, 49, 64, 6, 10, 16, 81, 8, 20, 15, 351, 229, 451, 59, 11, 7, 41, 19, 73, 92, 114, 27, 36, 48, 140, 12, 53, 17, 24, 33, 69, 86, 107, 44, 170, 18, 63, 22, 410, 28, 524, 76, 271, 101, 14, 23, 687, 529, 895, 253, 25, 97, 213, 145, 333, 3413, 67, 2091, 31, 607, 103, 415, 4531, 47, 131, 87, 193, 55

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

Compare to the formula for A246164. However, instead of reversing binary representation, we employ here balanced ternary enumeration of integers (see A117966).

Inverse: A246205.
Similar or related entanglement permutations: A246164, A245702, A246202, A246208, A246210.
Cf. A014580, A091242, A117966.

a(1) = 1, and for n > 1, if A117966(n) < 0, then a(n) = A014580(a(-(A117966(n)))), otherwise a(n) = A091242(a(A117966(n)-1)).
As a composition of related permutations:
a(n) = A245702(A246208(n)).
a(n) = A246202(A246210(n)).

cp's OEIS Frontend

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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A246162 Permutation of natural numbers: a(1) = 1, a(A000069(n)) = A014580(a(n-1)), a(A001969(n)) = A091242(a(n-1)), 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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A091246 Inverse function of A091242: position in A091242 if the n-th GF(2)[X] polynomial is reducible, 0 otherwise.

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A245813 Permutation of natural numbers induced when A091205 is restricted to {1} and binary codes for polynomials reducible over GF(2): a(1) = 1, a(n) = A062298(A091205(A091242(n-1))).

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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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A245820 Permutation of natural numbers induced when A245704 is restricted to {1} and binary codes for polynomials reducible over GF(2): a(1) = 1, a(n) = A062298(A245704(A091242(n-1))).

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