A305422 -id:A305422 - cp's OEIS frontend

A305418 Permutation of nonnegative integers: a(1) = 0, a(2n) = 1 + 2a(n), a(2n+1) = 2a(A305422(2n+1)).

0, 1, 2, 3, 6, 5, 4, 7, 10, 13, 8, 11, 16, 9, 14, 15, 30, 21, 32, 27, 12, 17, 34, 23, 64, 33, 22, 19, 18, 29, 128, 31, 258, 61, 36, 43, 256, 65, 38, 55, 512, 25, 130, 35, 46, 69, 1024, 47, 20, 129, 62, 67, 66, 45, 2048, 39, 70, 37, 4096, 59, 8192, 257, 26, 63, 54, 517, 16384, 123, 24, 73, 16386, 87, 32768, 513, 142, 131, 8194, 77, 132, 111, 48, 1025, 42, 51

Offset: 1

Antti Karttunen, Jun 10 2018

nonn

This is GF(2)[X] analog of A156552. Note the indexing: the domain starts from 1, while the range includes also zero.

Cf. A305417 (inverse).
Cf. A305422.
Cf. also A091203, A156552, A052331, A305428.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A305418(n) = if(1==n,(n-1),if(!(n%2),1+(2*(A305418(n/2))),2*A305418(A305422(n))));

a(1) = 0, a(2n) = 1 + 2*a(n), a(2n+1) = 2*a(A305422(2n+1)).
a(n) = A054429(A305428(n)).
For all n >= 1:
A000120(a(n)) = A091222(n).
A069010(a(n)) = A091221(n).
A106737(a(n)) = A091220(n).
A132971(a(n)) = A091219(n).
A085357(a(n)) = A304109(n).

A304529 a(1) = 0, a(2n) = n, a(2n+1) = a(A305422(2n+1)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 10 2018

nonn

This is GF(2)[X] analog of A246277.
For all i, j: a(i) = a(j) => A278233(i) = A278233(j).
For all i, j: a(i) = a(j) => A305788(i) = A305788(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A305422, A305425.
Cf. A014580 (positions of 1's), A278233, A305788.
Cf. also A246277.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A304529(n) = if(1==n,0,while(n%2, n = A305422(n)); n/2);

a(1) = 0, a(2n) = n, a(2n+1) = a(A305422(2n+1)).

A305425 a(n) = n/2 for even n, a(n) = A305422(n) for odd n.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 3, 4, 6, 5, 7, 6, 11, 7, 8, 8, 16, 9, 13, 10, 5, 11, 22, 12, 19, 13, 12, 14, 14, 15, 25, 16, 50, 17, 29, 18, 31, 19, 28, 20, 37, 21, 38, 22, 24, 23, 41, 24, 9, 25, 32, 26, 26, 27, 47, 28, 44, 29, 55, 30, 59, 31, 10, 32, 20, 33, 61, 34, 21, 35, 118, 36, 67, 37, 88, 38, 110, 39, 53, 40, 69, 41, 18, 42, 64, 43, 73, 44, 94, 45, 87, 46, 43, 47

Offset: 1

Antti Karttunen, Jun 08 2018

nonn

Each k occurs exactly twice, at 2k and at A305421(k).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to polynomials in ring GF(2)[X]

Bisections: A000027 and A305422.

Cf. also A252463.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A305425(n) = if(n%2,A305422(n),n/2);

a(n) = n/2 if n is even, a(n) = A305422(n) if n is odd.

A305428 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A305422(2n+1)).

Original entry on oeis.org

0, 1, 3, 2, 5, 6, 7, 4, 13, 10, 15, 12, 31, 14, 9, 8, 17, 26, 63, 20, 11, 30, 61, 24, 127, 62, 25, 28, 29, 18, 255, 16, 509, 34, 59, 52, 511, 126, 57, 40, 1023, 22, 253, 60, 49, 122, 2047, 48, 27, 254, 33, 124, 125, 50, 4095, 56, 121, 58, 8191, 36, 16383, 510, 21, 32, 41, 1018, 32767, 68, 23, 118, 32765, 104, 65535, 1022, 241, 252, 16381, 114, 251, 80, 47

Offset: 1

Antti Karttunen, Jun 10 2018

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

This is GF(2)[X] analog of A243071.

Cf. A305427 (inverse).
Cf. A305422.
Cf. also A243071, A305418.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A305428(n) = if(n<=2,(n-1),if(!(n%2),2*A305428(n/2),1+(2*(A305428(A305422(n))))));

a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A305422(2n+1)).

a(n) = A054429(A305418(n)).

A305424 Permutation of natural numbers: a(n) = A305422(2*n-1).

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 11, 8, 16, 13, 5, 22, 19, 12, 14, 25, 50, 29, 31, 28, 37, 38, 24, 41, 9, 32, 26, 47, 44, 55, 59, 10, 20, 61, 21, 118, 67, 88, 110, 53, 69, 18, 64, 73, 94, 87, 43, 52, 91, 100, 58, 97, 56, 15, 103, 62, 82, 109, 115, 48, 23, 74, 76, 49, 98, 117, 113, 152, 131, 46, 148, 137, 143, 164, 218, 27, 96, 227, 145, 230, 89, 182, 200

Offset: 1

Antti Karttunen, Jun 08 2018

nonn

Odd bisection of A305422 and A305425.

Cf. A305423 (inverse).
Cf. A014580, A091225, A304522, A305425.
Cf. also A064216.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A305424(n) = A305422(n+n-1);

a(n) = A305422(2*n-1).

A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.

Original entry on oeis.org

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

Offset: 1

David Petry (petry(AT)accessone.com)

nonn

Or, binary irreducible polynomials, interpreted as binary vectors, then written in base 10.
The numbers {a(n)} are a subset of the set {A206074}. - Thomas Ordowski, Feb 21 2014
2^n - 1 is a term if and only if n = 2 or n is a prime and 2 is a primitive root modulo n. - Jianing Song, May 10 2021
For odd k, k is a term if and only if binary_reverse(k) = A145341((k+1)/2) is. - Joerg Arndt and Jianing Song, May 10 2021

			x^4 + x^3 + 1 -> 16+8+1 = 25. Or, x^4 + x^3 + 1 -> 11001 (binary) = 25 (decimal).

A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (first 1377 terms from T. D. Noe)
Index entries for sequences operating on GF(2)[X]-polynomials

Written in binary: A058943.
Number of degree-n irreducible polynomials: A001037, see also A000031.
Multiplication table: A048720.
Characteristic function: A091225. Inverse: A091227. a(n) = A091202(A000040(n)). Almost complement of A091242. Union of A091206 & A091214 and also of A091250 & A091252. First differences: A091223. Apart from a(1) and a(2), a subsequence of A092246 and hence A000069.
Table of irreducible factors of n: A256170.
Irreducible polynomials satisfying particular conditions: A071642, A132447, A132449, A132453, A162570.
Factorization sentinel: A278239.
Sequences analyzing the difference between factorization into GF(2)[X] irreducibles and ordinary prime factorization of the corresponding integer: A234741, A234742, A235032, A235033, A235034, A235035, A235040, A236850, A325386, A325559, A325560, A325563, A325641, A325642, A325643.
Factorization-preserving isomorphisms: A091203, A091204, A235041, A235042.
See A115871 for sequences related to cross-domain congruences.
Functions based on the irreducibles: A305421, A305422.

fQ[n_] := Block[{ply = Plus @@ (Reverse@ IntegerDigits[n, 2] x^Range[0, Floor@ Log2@ n])}, ply == Factor[ply, Modulus -> 2] && n != 2^Floor@ Log2@ n]; fQ[2] = True; Select[ Range@ 378, fQ] (* Robert G. Wilson v, Aug 12 2011 *)
Reap[Do[If[IrreduciblePolynomialQ[IntegerDigits[n, 2] . x^Reverse[Range[0, Floor[Log[2, n]]]], Modulus -> 2], Sow[n]], {n, 2, 1000}]][[2, 1]] (* Jean-François Alcover, Nov 21 2016 *)

is(n)=polisirreducible(Pol(binary(n))*Mod(1,2)) \\ Charles R Greathouse IV, Mar 22 2013

A091203 Factorization-preserving isomorphism from binary codes of GF(2) polynomials to integers.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 11, 10, 27, 16, 81, 30, 13, 36, 25, 14, 33, 24, 17, 22, 45, 20, 21, 54, 19, 32, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 48, 35, 34, 243, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47, 324

Offset: 0

Antti Karttunen, Jan 03 2004

nonn

E.g. we have the following identities: A000040(n) = a(A014580(n)), A091219(n) = A008683(a(n)), A091220(n) = A000005(a(n)), A091221(n) = A001221(a(n)), A091222(n) = A001222(a(n)), A091225(n) = A010051(a(n)), A091227(n) = A049084(a(n)), A091247(n) = A066247(a(n)).

Antti Karttunen, Table of n, a(n) for n = 0..8192
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials
Index entries for sequences that are permutations of the natural numbers

Inverse: A091202.
Several variants exist: A235042, A091205, A106443, A106445, A106447.
Cf. also A000005, A000040, A001221, A001222, A004247, A008683, A010051, A014580, A048720, A049084, A066247, A091219, A091220, A091221, A091222, A091225, A091227, A091247, A234741, A234742, A245703, A245704, A278233.
Cf. also A302024, A302026, A305418, A305428 for other similar permutations.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A091203(n) = if(n<=1,n,if(!(n%2),2*A091203(n/2),A003961(A091203(A305422(n))))); \\ Antti Karttunen, Jun 10 2018

a(0)=0, a(1)=1. For n's coding an irreducible polynomial ir_i, that is if n=A014580(i), we have a(n) = A000040(i) and for composite polynomials a(ir_i X ir_j X ...) = p_i * p_j * ..., where p_i = A000040(i) and X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and * for the ordinary multiplication of integers (A004247).
Other identities. For all n >= 1, the following holds:
A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials, A014580, to primes and the binary representations of corresponding reducible polynomials, A091242, to composite numbers. The permutations A091205, A106443, A106445, A106447, A235042 and A245704 have the same property.]
From Antti Karttunen, Jun 10 2018: (Start)
For n <= 1, a(n) = n, for n > 1, a(n) = 2*a(n/2) if n is even, and if n is odd, then a(n) = A003961(a(A305422(n))).
a(n) = A005940(1+A305418(n)) = A163511(A305428(n)).
A046523(a(n)) = A278233(n).
(End)

A305419 Largest k < n whose binary expansion encodes an irreducible (0,1)-polynomial over GF(2)[X], with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 3, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 13, 13, 19, 19, 19, 19, 19, 19, 25, 25, 25, 25, 25, 25, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 41, 41, 41, 41, 47, 47, 47, 47, 47, 47, 47, 47, 55, 55, 55, 55, 59, 59, 61, 61, 61, 61, 61, 61, 67, 67, 67, 67, 67, 67, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 87

Offset: 1

Antti Karttunen, Jun 07 2018

nonn

For n >= 3, a(n) is the largest term of A014580 less than n.

Antti Karttunen, Table of n, a(n) for n = 1..21845
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A014580, A091225, A305420, A305422.

Cf. also A151799 (A136548), A305429.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));

A305421 GF(2)[X] factorization prime shift towards larger terms.

Original entry on oeis.org

1, 3, 7, 5, 21, 9, 11, 15, 49, 63, 13, 27, 19, 29, 107, 17, 273, 83, 25, 65, 69, 23, 121, 45, 31, 53, 151, 39, 35, 189, 37, 51, 251, 819, 173, 245, 41, 43, 233, 195, 47, 207, 93, 57, 997, 139, 55, 119, 127, 33, 1911, 95, 79, 441, 59, 105, 367, 101, 61, 455, 67, 111, 475, 85, 1281, 269, 73, 1365, 81, 503, 457, 287, 87, 123, 1549, 125, 179, 315

Offset: 1

Antti Karttunen, Jun 07 2018

nonn

Permutation of the odd numbers, A005408.

Let a x b stand for the carryless binary multiplication of positive integers a and b, that is, the result of operation A048720(a,b). With n having a unique factorization as A014580(i) x A014580(j) x ... x A014580(k), 1 <= i <= j <= ... <= k, a(n) = A014580(1+i) x A014580(1+j) x ... x A014580(1+k).

			For n = 12, which by its binary representation '1100' corresponds with (0,1)-polynomial x^3 + x^2, which over GF(2)[X] is factored as (x)(x)(x+1), i.e., 12 = A048720(2,A048720(2,3)) = A048720(A014580(1), A048720(A014580(1),A014580(2))), we then form a(12) as A048720(A014580(2), A048720(A014580(2),A014580(3))) = A048720(3,A048720(3,7)) = 27. Note that x, x+1 and x^2 + x + 1 are the three smallest irreducible (0,1)-polynomials when factored over GF(2)[X], and their binary representations 2, 3 and 7 are the three initial terms of A014580.

Antti Karttunen, Table of n, a(n) for n = 1..21845
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A305422 (a left inverse).
Cf. A001317, A014580, A091225, A193231, A268389, A305420, A305423.
Cf. also A003961, A300841.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305420(n) = { my(k=1+n); while(!A091225(k),k++); (k); };
A305421(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305420(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };

For all n >= 1:
A305422(a(n)) = n.
A268389(a(n)) = A007814(n).
a(A000079(n)) = A001317(n).

cp's OEIS Frontend

A305418 Permutation of nonnegative integers: a(1) = 0, a(2n) = 1 + 2*a(n), a(2n+1) = 2*a(A305422(2n+1)).

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A304529 a(1) = 0, a(2n) = n, a(2n+1) = a(A305422(2n+1)).

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A305425 a(n) = n/2 for even n, a(n) = A305422(n) for odd n.

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A305428 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A305422(2n+1)).

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A305424 Permutation of natural numbers: a(n) = A305422(2*n-1).

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A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.

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A091203 Factorization-preserving isomorphism from binary codes of GF(2) polynomials to integers.

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A305419 Largest k < n whose binary expansion encodes an irreducible (0,1)-polynomial over GF(2)[X], with a(1) = a(2) = 1.

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A305418 Permutation of nonnegative integers: a(1) = 0, a(2n) = 1 + 2a(n), a(2n+1) = 2a(A305422(2n+1)).

A305428 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A305422(2n+1)).