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 1-6 of 6 results.

A091205 Factorization and index-recursion 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, 23, 10, 27, 16, 81, 30, 13, 36, 25, 14, 69, 24, 11, 46, 45, 20, 21, 54, 19, 32, 57, 162, 115, 60, 47, 26, 63, 72, 61, 50, 33, 28, 135, 138, 17, 48, 35, 22, 243, 92, 39, 90, 37, 40, 207, 42, 83, 108, 29, 38, 75, 64, 225, 114, 103
Offset: 0

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Comments

This "deeply multiplicative" bijection is one of the deep variants of A091203 which satisfy most of the same identities as the latter, but it additionally preserves also the structures where we recurse on irreducible polynomial's A014580-index. E.g., we have: A091238(n) = A061775(a(n)). The reason this holds is that when the permutation is restricted to the binary codes for irreducible polynomials over GF(2) (A014580), it induces itself: a(n) = A049084(a(A014580(n))).
On the other hand, when this permutation is restricted to the union of {1} and reducible polynomials over GF(2) (A091242), permutation A245813 is induced.

Links

Crossrefs

Inverse: A091204.
Similar or related permutations: A091203, A106443, A106445, A106447, A235042, A245704, A245813, A245821.
Cf. A000040, A007097, A010051, A014580, A049084, A061775, A091238, A091225, A091226, A091230, A091242.

Programs

  • PARI
    allocatemem(123456789);
v091226 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
n=2; while((n < 2^22), if(isA014580(n), v091226[n] = v091226[n-1]+1, v091226[n] = v091226[n-1]); n++)
A091226(n) = v091226[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)}));
for(n=0, 8192, write("b091205.txt", n, " ", A091205(n)));
\\ Antti Karttunen, Aug 16 2014

Formula

a(0)=0, a(1)=1. For n that is coding an irreducible polynomial, that is if n = A014580(i), we have a(n) = A000040(a(i)) and for reducible polynomials a(ir_i X ir_j X ...) = a(ir_i) * a(ir_j) * ..., where ir_i = A014580(i), X stands for carryless multiplication of polynomials over GF(2) (A048720) and * for the ordinary multiplication of integers (A004247).
As a composition of related permutations:
a(n) = A245821(A245704(n)).
Other identities.
For all n >= 0, the following holds:
a(A091230(n)) = A007097(n). [Maps iterates of A014580 to the iterates of primes. Permutation A245704 has the same property.]
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, bijectively to primes and the binary representations of corresponding reducible polynomials, A091242, to composite numbers, in some order. The permutations A091203, A106443, A106445, A106447, A235042 and A245704 have the same property.]

Extensions

Name changed by Antti Karttunen, Aug 16 2014

A245703 Permutation of natural numbers: a(1) = 1, a(p_n) = A014580(a(n)), a(c_n) = A091242(a(n)), where p_n = n-th prime, c_n = n-th composite number and A014580(n) and A091242(n) are binary codes for n-th irreducible and n-th reducible polynomials over GF(2), respectively.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 11, 6, 8, 12, 25, 9, 13, 17, 10, 14, 47, 18, 19, 34, 15, 20, 31, 24, 16, 21, 62, 26, 55, 27, 137, 45, 22, 28, 42, 33, 37, 23, 29, 79, 59, 35, 87, 71, 36, 166, 41, 58, 30, 38, 54, 44, 61, 49, 32, 39, 99, 76, 319, 46, 91, 108, 89, 48, 200, 53, 97, 75, 40, 50, 203, 70, 67, 57, 78, 64, 43, 51
Offset: 1

Views

Author

Antti Karttunen, Aug 02 2014

Keywords

Comments

All the permutations A091202, A091204, A106442, A106444, A106446, A235041 share the same property that primes (A000040) are mapped bijectively to the binary representations of irreducible GF(2) polynomials (A014580) but while they determine the mapping of composites (A002808) to the corresponding binary codes of reducible polynomials (A091242) by a simple multiplicative rule, this permutation employs index-recursion also in that case.

Links

Crossrefs

Inverse: A245704.
Cf. A000040, A002808, A000720, A007097, A010051, A014580, A065855, A091225, A091230, A091242.
Similar or related permutations: A091202, A091204, A106442, A106444, A106446, A235041, A135141, A245701, A245702, A245821, A245822, A244987, A245450.

Programs

  • PARI
    allocatemem(123456789);
a014580 = vector(2^18);
a091242 = 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++; a014580[i] = n, j++; a091242[j] = n); n++)
A245703(n) = if(1==n, 1, if(isprime(n), a014580[A245703(primepi(n))], a091242[A245703(n-primepi(n)-1)]));
for(n=1, 10001, write("b245703.txt", n, " ", A245703(n)));
  • Scheme
    ;; With memoization-macro definec.
(definec (A245703 n) (cond ((= 1 n) n) ((= 1 (A010051 n)) (A014580 (A245703 (A000720 n)))) (else (A091242 (A245703 (A065855 n))))))

Formula

a(1) = 1, a(p_n) = A014580(a(n)) and a(c_n) = A091242(a(n)), where p_n is the n-th prime, A000040(n) and c_n is the n-th composite, A002808(n).
a(1) = 1, after which, if A010051(n) is 1 [i.e. n is prime], then a(n) = A014580(a(A000720(n))), otherwise a(n) = A091242(a(A065855(n))).
As a composition of related permutations:
a(n) = A245702(A135141(n)).
a(n) = A091204(A245821(n)).
Other identities. For all n >= 1, the following holds:
a(A007097(n)) = A091230(n). [Maps iterates of primes to the iterates of A014580. Permutation A091204 has the same property]
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 A091202, A091204, A106442, A106444, A106446 and A235041 have the same property.]

A245704 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A000040(a(n)), a(A091242(n)) = A002808(a(n)), where A000040(n) = n-th prime, A002808(n) = n-th composite number, and A014580(n) and A091242(n) are binary codes for n-th irreducible and n-th reducible polynomial over GF(2), respectively.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 15, 7, 10, 13, 16, 21, 25, 14, 18, 19, 22, 26, 33, 38, 24, 11, 28, 30, 34, 39, 49, 23, 55, 36, 20, 42, 45, 37, 50, 56, 69, 47, 35, 77, 52, 32, 60, 17, 64, 54, 70, 78, 94, 66, 51, 29, 105, 74, 48, 41, 84, 53, 27, 88, 76, 95, 106, 73, 125, 91, 72, 44, 140, 97, 100, 68, 58, 115, 75, 40
Offset: 1

Views

Author

Antti Karttunen, Aug 02 2014

Keywords

Comments

All the permutations A091203, A091205, A106443, A106445, A106447, A235042 share the same property that the binary representations of irreducible GF(2) polynomials (A014580) are mapped bijectively to the primes (A000040) but while they determine the mapping of corresponding reducible polynomials (A091242) to the composite numbers (A002808) by a simple multiplicative rule, this permutation employs index-recursion also in that case.

Links

Crossrefs

Inverse: A245703.
Cf. A000040, A002808, A007097, A010051, A014580, A091225, A091226, A091230, A091242, A091245.
Similar or related permutations: A245701, A245822, A227413, A091203, A091205, A106443, A106445, A106447, A235042, A244987, A245450.

Programs

Formula

a(1) = 1, after which, if A091225(n) is 1 [i.e. n is in A014580], then a(n) = A000040(a(A091226(n))), otherwise a(n) = A002808(a(A091245(n))).
As a composition of related permutations:
a(n) = A227413(A245701(n)).
a(n) = A245822(A091205(n)).
Other identities. For all n >= 1, the following holds:
a(A091230(n)) = A007097(n). [Maps iterates of A014580 to the iterates of primes. Permutation A091205 has the same property].
A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials (= A014580) to primes and the corresponding representations of reducible polynomials to composites].

A091204 Factorization and index-recursion preserving isomorphism from nonnegative integers to polynomials over GF(2).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 25, 12, 19, 22, 9, 16, 47, 10, 31, 28, 29, 50, 13, 24, 21, 38, 15, 44, 61, 18, 137, 32, 43, 94, 49, 20, 55, 62, 53, 56, 97, 58, 115, 100, 27, 26, 37, 48, 69, 42, 113, 76, 73, 30, 79, 88, 33, 122, 319, 36, 41, 274, 39, 64, 121, 86, 185
Offset: 0

Views

Author

Antti Karttunen, Jan 03 2004. Name changed Aug 16 2014

Keywords

Comments

This "deeply multiplicative" isomorphism is one of the deep variants of A091202 which satisfies most of the same identities as the latter, but it additionally preserves also the structures where we recurse on prime's index. E.g. we have: A091230(n) = a(A007097(n)) and A061775(n) = A091238(a(n)). This is because the permutation induces itself when it is restricted to the primes: a(n) = A091227(a(A000040(n))).
On the other hand, when this permutation is restricted to the nonprime numbers (A018252), permutation A245814 is induced.

Links

Crossrefs

Inverse: A091205.
Similar or related permutations: A091202, A106442, A106444, A106446, A235041, A245703, A245814, A245822.
Cf. A000040, A007097, A010051, A014580, A018252, A048720, A061775, A091225, A091230, A091238, A091242.

Programs

  • PARI
    v014580 = vector(2^18); A014580(n) = v014580[n];
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n); n++)
A091204(n) = if(n<=1, n, if(isprime(n), A014580(A091204(primepi(n))), {my(pfs, t, bits, i); pfs=factor(n); pfs[,1]=apply(t->Pol(binary(A091204(t))), pfs[,1]); sum(i=1, #bits=Vec(factorback(pfs))%2, bits[i]<<(#bits-i))}));
for(n=0, 8192, write("b091204.txt", n, " ", A091204(n)));
\\ Antti Karttunen, Aug 16 2014

Formula

a(0)=0, a(1)=1, a(p_i) = A014580(a(i)) for primes 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).
As a composition of related permutations:
a(n) = A245703(A245822(n)).
Other identities.
For all n >= 0, the following holds:
a(A007097(n)) = A091230(n). [Maps iterates of primes to the iterates of A014580. Permutation A245703 has the same property]
For all n >= 1, the following holds:
A091225(a(n)) = A010051(n). [Maps primes bijectively to binary representations of irreducible GF(2) polynomials, A014580, and nonprimes to union of {1} and the binary representations of corresponding reducible polynomials, A091242, in some order. The permutations A091202, A106442, A106444, A106446, A235041 and A245703 have the same property.]

A091238 Number of nodes in rooted tree with GF2X-Matula number n.

Original entry on oeis.org

1, 2, 3, 3, 5, 4, 4, 4, 6, 6, 4, 5, 6, 5, 7, 5, 9, 7, 5, 7, 7, 5, 8, 6, 5, 7, 8, 6, 6, 8, 5, 6, 7, 10, 9, 8, 7, 6, 8, 8, 7, 8, 7, 6, 10, 9, 5, 7, 7, 6, 11, 8, 7, 9, 6, 7, 10, 7, 7, 9, 6, 6, 9, 7, 11, 8, 8, 11, 7, 10, 8, 9, 6, 8, 12, 7, 9, 9, 8, 9, 11, 8, 9, 9, 13, 8, 10, 7, 8, 11, 8, 10, 8, 6, 9, 8
Offset: 1

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Comments

Each n occurs A000081(n) times.

Examples

			GF2X-Matula numbers for unoriented rooted trees are constructed otherwise just like the standard Matula-Goebel numbers (cf. A061773), but instead of normal factorization in N, one factorizes in polynomial ring GF(2)[X] as follows. Here IR(n) is the n-th irreducible polynomial (A014580(n)) and X stands for GF(2)[X]-multiplication (A048720):
................................................o...................o
................................................|...................|
............o...............o...o........o......o...............o...o
............|...............|...|........|......|...............|...|
...o........o......o...o....o...o....o...o......o......o.o.o....o...o
...|........|.......\./......\./......\./.......|.......\|/......\./.
x..x........x........x........x........x........x........x........x..
1..2 = IR(1)..3 = IR(2)..4 = 2 X 2....5 = 3 X 3....6 = 2 X 3....7 = IR(3)..8 = 2 X 2 X 2..9 = 3 X 7
Counting the vertices (marked with x's and o's) of each tree above, we get the eight initial terms of this sequence: 1,2,3,3,5,4,4,4,6.

Links

Crossrefs

a(n) = A061775(A091205(n)). a(A091230(n)) = n+1. Cf. A091239-A091241.

A091240 Largest GF2X-Matula number i which encodes a tree of n nodes, i.e., for which A091238(i) = n.

Original entry on oeis.org

1, 2, 4, 11, 47, 319, 3053, 40345
Offset: 1

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Comments

Apparently from n=4 onward given by recurrence a(4) = A014580(4), a(5) = A014580(A014580(4)), a(6) = A014580(A014580(A014580(4))), etc.

Links

Crossrefs

Analogous to A005518. A091239 gives the smallest i with number of nodes = n. Cf. A091230, A091238, A091241.
Showing 1-6 of 6 results.

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