cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A091202 Factorization-preserving isomorphism from nonnegative integers to binary codes for polynomials over GF(2).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 13, 12, 19, 22, 9, 16, 25, 10, 31, 28, 29, 26, 37, 24, 21, 38, 15, 44, 41, 18, 47, 32, 23, 50, 49, 20, 55, 62, 53, 56, 59, 58, 61, 52, 27, 74, 67, 48, 69, 42, 43, 76, 73, 30, 35, 88, 33, 82, 87, 36, 91, 94, 39, 64, 121, 46, 97, 100, 111, 98
Offset: 0

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Comments

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

Links

Crossrefs

Inverse: A091203.
Several variants exist: A235041, A091204, A106442, A106444, A106446.
Cf. also A000005, A091220, A001221, A091221, A001222, A091222, A008683, A091219, A000040, A014580, A048720, A049084, A091227, A245703, A234741.
Cf. also A302023, A302025, A305417, A305427 for other similar permutations.

Programs

  • PARI
    A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
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); };
A091202(n) = if(n<=1,n,if(!(n%2),2*A091202(n/2),A305421(A091202(A064989(n))))); \\ Antti Karttunen, Jun 10 2018

Formula

a(0)=0, a(1)=1, a(p_i) = A014580(i) for primes p_i with index i and for composites a(p_i * p_j * ...) = a(p_i) X a(p_j) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720).
Other identities. For all n >= 1, the following holds:
A091225(a(n)) = A010051(n). [Maps primes to binary representations of irreducible GF(2) polynomials, A014580, and nonprimes to union of {1} and the binary representations of corresponding reducible polynomials, A091242. The permutations A091204, A106442, A106444, A106446, A235041 and A245703 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) = A305421(a(A064989(n))).
a(n) = A305417(A156552(n)) = A305427(A243071(n)).
(End)

A305417 Permutation of natural numbers: a(0) = 1, a(2n) = A305421(a(n)), a(2n+1) = 2*a(n).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 11, 14, 9, 12, 21, 10, 15, 16, 13, 22, 29, 28, 49, 18, 27, 24, 69, 42, 63, 20, 107, 30, 17, 32, 19, 26, 23, 44, 35, 58, 39, 56, 127, 98, 83, 36, 151, 54, 45, 48, 81, 138, 207, 84, 475, 126, 65, 40, 743, 214, 189, 60, 273, 34, 51, 64, 25, 38, 53, 52, 121, 46, 57, 88, 173, 70, 101, 116, 233, 78, 105, 112, 199, 254, 129
Offset: 0

Views

Author

Antti Karttunen, Jun 10 2018

Keywords

Comments

This is GF(2)[X] analog of A005940, but note the indexing: here the domain starts from 0, although the range excludes zero.
This sequence can be represented as a binary tree. Each child to the left is obtained by applying A305421 to the parent, and each child to the right is obtained by doubling the parent:
1
|
...................2...................
3 4
7......../ \........6 5......../ \........8
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
11 14 9 12 21 10 15 16
13 22 29 28 49 18 27 24 69 42 63 20 107 30 17 32
Sequence A305427 is obtained by scanning the same tree level by level from right to left.

Links

Crossrefs

Cf. A305418 (inverse), A305427 (mirror image).
Cf. A014580 (left edge from 2 onward), A305421.
Cf. also A005940, A052330, A091202.

Programs

  • PARI
    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); };
A305417(n) = if(0==n,(1+n),if(!(n%2),A305421(A305417(n/2)),2*(A305417((n-1)/2))));

Formula

a(0) = 1, a(2n) = A305421(a(n)), a(2n+1) = 2*a(n).
a(n) = A305427(A054429(n)).
For all n >= 1, a(A000079(n-1)) = A014580(n).

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)).

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

Views

Author

Antti Karttunen, Jun 10 2018

Keywords

Comments

Note the indexing: the domain starts from 1, while the range includes also zero.
This is GF(2)[X] analog of A243071.

Links

Crossrefs

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

Programs

  • PARI
    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))))));

Formula

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)).
Showing 1-3 of 3 results.

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