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.

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A075166 Natural numbers mapped to Dyck path encodings of the rooted plane trees obtained by recursing on the exponents of the prime factorization of n.

Original entry on oeis.org

0, 10, 1010, 1100, 101010, 101100, 10101010, 110100, 110010, 10101100, 1010101010, 10110100, 101010101010, 1010101100, 10110010, 111000, 10101010101010, 11001100, 1010101010101010, 1010110100, 1010110010
Offset: 1

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Author

Antti Karttunen, Sep 13 2002

Keywords

Comments

Note that we recurse on the exponent + 1 for all other primes except the largest one in the factorization. Thus for 6 = 3^1 * 2^1 we construct a tree by joining trees 1 and 2 with a new root node, for 7 = 7^1 * 5^0 * 3^0 * 2^0 we join four 1-trees (single leaves) with a new root node, for 8 = 2^3 we add a single edge below tree 3 and for 9 = 3^2 * 2^0 we join trees 2 and 1, to get the mirror image of tree 6. Compare to Matula/Goebel numbering of (unoriented) rooted trees as explained in A061773.

Examples

			The rooted plane trees encoded here are:
.....................o...............o.........o...o..o.......
.....................|...............|..........\./...|.......
.......o....o...o....o....o.o.o..o...o.o.o.o.o...o....o...o...
.......|.....\./.....|.....\|/....\./...\|.|/....|.....\./....
*......*......*......*......*......*......*......*......*.....
1......2......3......4......5......6......7......8......9.....

Links

Crossrefs

Permutation of A063171. Same sequence shown in decimal: A075165. The digital length of each term / 2 (the number of o-nodes in the corresponding trees) is given by A075167. Cf. A075171, A007088.

Formula

a(n) = A007088(A075165(n)) = A106456(A106442(n)). - Antti Karttunen, May 09 2005

A091227 Inverse function of A014580: position in A014580 if the n-th GF(2)[X] polynomial is irreducible, 0 otherwise.

Original entry on oeis.org

0, 1, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0, 5, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 9, 0, 0, 0, 10, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 13, 0, 14, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 18, 0, 0, 0, 0, 0, 19, 0
Offset: 1

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Links

Crossrefs

Inverse of A014580. a(n) = A049084(A091203(n)).

Formula

a(n) = A091225(n) * A091226(n).

A106442 Exponent-recursed cross-domain bijection from N to GF(2)[X]. Position of A075166(n) in A106456.

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, 128, 23, 50, 49, 20, 55, 62, 53, 56, 59, 58, 61, 52, 27, 74, 67, 192, 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, May 09 2005

Keywords

Comments

This map from the multiplicative domain of N to that of GF(2)[X] preserves Catalan-family structures, e.g. A106454(n) = a(A075164(n)), A075163(n) = A106453(a(n)), A075165(n) = A106455(a(n)), A075166(n) = A106456(a(n)), A075167(n) = A106457(a(n)). Shares with A091202 and A106444 the property that maps A000040(n) to A014580(n). Differs from the former for the first time at n=32, where A091202(32)=32, while a(32)=128. Differs from the latter for the first time at n=48, where A106444(48)=48, while a(48)=192.

Examples

			a(5) = 7, as 5 is the 3rd prime and the third irreducible GF(2)[X] polynomial x^2+x+1 is encoded as A014580(3) = 7. a(32) = a(2^5) = A048723(A014580(1),a(5)) = A048723(2,7) = 128. a(48) = a(3 * 2^4) = 3 X A048723(2,a(4+1)-1) = 3 X A048723(2,7-1) = 3 X 64 = 192.

Links

Crossrefs

Inverse: A106443. a(n) = A106454(A075163(n)).

Formula

a(0)=0, a(1)=1, a(p_i) = A014580(i) for primes p_i with index i and for composites n = p_i^e_i * p_j^e_j * p_k^e_k * ..., a(n) = A048723(a(p_i), a(e_i)) X A048723(a(p_j), a(1+e_j)-1) X A048723(a(p_k), a(1+e_k)-1) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and A048723(n, y) raises the n-th GF(2)[X] polynomial to the y:th power. Here p_i is the most significant prime in the factorization of n; its exponent e_i is not incremented before the recursion step, while the exponents of less significant primes e_j, e_k, ... are incremented by one before recursing and the result of the recursion is decremented by one before use.

A106443 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Position of A106456(n) in A075166.

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, 512, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 768, 35, 34, 19683, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47
Offset: 0

Views

Author

Antti Karttunen, May 09 2005

Keywords

Comments

This map from the multiplicative domain of GF(2)[X] to that of N preserves Catalan-family structures, e.g. A075164(n) = a(A106454(n)), A106453(n) = A075163(a(n)), A106455(n) = A075165(a(n)), A106456(n) = A075166(a(n)), A106457(n) = A075167(a(n)). Shares with A091203 and A106445 the property that maps A014580(n) to A000040(n). Differs from the former for the first time at n=32, where A091203(32)=32, while a(32)=512. Differs from the latter for the first time at n=48, where A106445(48)=48, while a(48)=768.

Examples

			a(5) = 9, as 5 encodes the GF(2)[X] polynomial x^2+1, which is the square of the second irreducible GF(2)[X] polynomial x+1 (encoded as 3) and the square of the second prime is 3^2=9. a(32) = a(A048723(2,5)) = 2^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = 3 * 2^(a(4+1)-1) = 3 * 2^(9-1) = 3 * 256 = 768.

Links

Crossrefs

Inverse: A106442. a(n) = A075164(A106453(n)).

Formula

a(0)=0, a(1)=1. For irreducible GF(2)[X] polynomials ir_i with index i (i.e. A014580(i)), a(ir_i) = A000040(i) and for composite polynomials n = A048723(ir_i, e_i) X A048723(ir_j, e_j) X A048723(ir_k, e_k) X ..., a(n) = a(ir_i)^a(e_i) * a(ir_j)^(a(1+e_j)-1) * a(ir_k)^(a(1+e_k)-1) * ... = A000040(i)^a(e_i) * A000040(j)^(a(1+e_j)-1) * A000040(k)^(a(1+e_k)-1), where X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and A048723(n, y) raises the n-th GF(2)[X] polynomial to the y:th power, while * is the ordinary multiplication and ^ is the ordinary exponentiation. Here ir_i is the most significant (largest) irreducible polynomial in the factorization of n; its exponent e_i is not incremented before the recursion step, while the exponents of less significant factors e_j, e_k, ... are incremented by one before recursing and the result of the recursion is decremented by one before use.

A106444 Exponent-recursed cross-domain bijection from N to GF(2)[X]. Variant of A091202 and A106442.

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, 128, 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
Offset: 0

Views

Author

Antti Karttunen, May 09 2005

Keywords

Comments

This map from the multiplicative domain of N to that of GF(2)[X] preserves 'superfactorized' structures, e.g. A106490(n) = A106493(a(n)), A106491(n) = A106494(a(n)), A064372(n) = A106495(a(n)). Shares with A091202 and A106442 the property that maps A000040(n) to A014580(n). Differs from A091202 for the first time at n=32, where A091202(32)=32, while a(32)=128. Differs from A106442 for the first time at n=48, where A106442(48)=192, while a(48)=48. Differs from A106446 for the first time at n=11, where A106446(11)=25, while a(11)=13.

Examples

			a(5) = 7, as 5 is the 3rd prime and the third irreducible GF(2)[X] polynomial x^2+x+1 is encoded as A014580(3) = 7. a(32) = a(2^5) = A048723(A014580(1),a(5)) = A048723(2,7) = 128. a(48) = a(3 * 2^4) = 3 X A048723(2,a(4)) = 3 X A048723(2,4) = 3 X 16 = 48.

Links

Crossrefs

Inverse: A106445.

Formula

a(0)=0, a(1)=1, a(p_i) = A014580(i) for primes p_i with index i and for composites n = p_i^e_i * p_j^e_j * p_k^e_k * ..., a(n) = A048723(a(p_i), a(e_i)) X A048723(a(p_j), a(e_j)) X A048723(a(p_k), a(e_k)) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and A048723(n, y) raises the n-th GF(2)[X] polynomial to the y:th power.

A106445 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Variant of A091203 and A106443.

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, 512, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 48, 35, 34, 19683, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47
Offset: 0

Views

Author

Antti Karttunen, May 09 2005

Keywords

Comments

This map from the multiplicative domain of GF(2)[X] to that of N preserves 'superfactorized' structures, e.g. A106493(n) = A106490(a(n)), A106494(n) = A106491(a(n)), A106495(n) = A064372(a(n)). Shares with A091203 and A106443 the property that maps A014580(n) to A000040(n). Differs from the plain variant A091203 for the first time at n=32, where A091203(32)=32, while a(32)=512. Differs from the variant A106443 for the first time at n=48, where A106443(48)=768, while a(48)=48. Differs from a yet deeper variant A106447 for the first time at n=13, where A106447(13)=23, while a(13)=11.

Examples

			a(5) = 9, as 5 encodes the GF(2)[X] polynomial x^2+1, which is the square of the second irreducible GF(2)[X] polynomial x+1 (encoded as 3) and the square of the second prime is 3^2=9. a(32) = a(A048723(2,5)) = 2^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = 3 * 2^a(4) = 3 * 2^4 = 3 * 16 = 48.

Links

Crossrefs

Inverse: A106444.

Formula

a(0)=0, a(1)=1. For irreducible GF(2)[X] polynomials ir_i with index i (i.e. A014580(i)), a(ir_i) = A000040(i) and for composite polynomials n = A048723(ir_i, e_i) X A048723(ir_j, e_j) X A048723(ir_k, e_k) X ..., a(n) = a(ir_i)^a(e_i) * a(ir_j)^a(e_j) * a(ir_k)^a(e_k) * ... = A000040(i)^a(e_i) * A000040(j)^a(e_j) * A000040(k)^a(e_k), where X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and A048723(n, y) raises the n-th GF(2)[X] polynomial to the y:th power, while * is the ordinary multiplication and ^ is the ordinary exponentiation.

A091219 Moebius-analog for the domain GF(2)[X]: a(n)=0 if A091221(n)!=A091222(n) (i.e., if the polynomial is not squarefree), otherwise (-1)^A091222(n).

Original entry on oeis.org

1, -1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 0, -1, 1, 0, 0, 0, -1, -1, 0, 0, 1, 1, 0, -1, 1, 0, 0, 1, 0, -1, 0, 1, 0, 1, 0, -1, 1, 0, 0, -1, 0, 1, 0, 0, -1, -1, 0, 1, 1, 0, 0, 1, 0, -1, 0, 0, -1, -1, 0, -1, 1, 0, 0, 0, -1, -1, 0, 0, -1, 1, 0, -1, 1, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, -1, -1, 0, 1, 0
Offset: 1

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Comments

The absolute values give a characteristic function for squarefree GF(2)[X]-polynomials.

Links

Crossrefs

a(n) = A008683(A091203(n)) = A008683(A091205(n)).

A106456 Natural numbers mapped to Dyck path encodings of the rooted plane trees obtained by recursing on the exponents of the GF(2)[X] factorization of n.

Original entry on oeis.org

0, 10, 1010, 1100, 110010, 101100, 101010, 110100, 10110010, 11001100, 10101010, 10110100, 1010101010, 10101100, 11010010, 111000, 11100010, 1011001100, 101010101010, 1100110100, 11001010, 1010101100, 101010110010
Offset: 1

Views

Author

Antti Karttunen, May 09 2005

Keywords

Comments

Note that we recurse on the exponent + 1 for all other irreducible polynomials except the largest one in the GF(2)[X] factorization. Thus for 6 = A048723(3,1) X A048723(2,1) we construct a tree by joining trees 1 and 2 with a new root node, for 7 = A048723(7,1) X A048723(3,0) X A048723(2,0) we join three 1-trees (single leaves) with a new root node, for 8 = A048273(2,3) we add a single edge below tree 3 and for 9 = A048723(7,1) X A048723(3,1) X A048273(2,0) we connect the trees 1 and 2 and 1 with a new root node.

Examples

			The rooted plane trees encoded here are:
.....................o....o..........o.........o...o....o.....
.....................|....|..........|..........\./.....|.....
.......o....o...o....o....o...o..o...o..o.o.o....o....o.o.o...
.......|.....\./.....|.....\./....\./....\|/.....|.....\|/....
*......*......*......*......*......*......*......*......*.....
1......2......3......4......5......6......7......8......9.....

Links

Crossrefs

a(n) = A007088(A106455(n)) = A075166(A106443(n)). GF(2)[X]-analog of A075166. Permutation of A063171. Same sequence shown in decimal: A106455. The digital length of each term / 2 (the number of o-nodes in the corresponding trees) is given by A106457. Cf. A106451-A106454.

A091257 Multiplication table A x B computed for polynomials over GF(2), where (A,B) runs as (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 5, 8, 5, 6, 10, 12, 12, 10, 6, 7, 12, 15, 16, 15, 12, 7, 8, 14, 10, 20, 20, 10, 14, 8, 9, 16, 9, 24, 17, 24, 9, 16, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 20, 27, 32, 27, 20, 27, 32, 27, 20, 11, 12, 22, 30, 36, 40, 18, 18, 40, 36, 30, 22, 12
Offset: 1

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Comments

Essentially same as A048720 but computed starting from offset one instead of zero. Analogous to A003991. Each n occurs A091220(n) times.

Links

Crossrefs

a(n) = A048720bi(A091255(n), A091256(n)) because the identity A x B = gcd(A, B) x lcm(A, B) holds also in the polynomial ring GF(2)[X].

A106455 Sequence A106456 interpreted as binary numbers and converted to decimal.

Original entry on oeis.org

0, 2, 10, 12, 50, 44, 42, 52, 178, 204, 170, 180, 682, 172, 210, 56, 226, 716, 2730, 820, 202, 684, 2738, 184, 10922, 2732, 722, 692, 690, 844, 43690, 228, 174770, 908, 2762, 2868, 174762, 10924, 2770, 824, 699050, 812, 43698, 2740, 738, 10956
Offset: 1

Views

Author

Antti Karttunen, May 09 2005

Keywords

Links

Crossrefs

a(n) = A075165(A106443(n)). Permutation of A014486. Same sequence shown in binary: A106456. The binary width of each term / 2 is given by A106457. GF(2)[X]-analog of A075165.
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