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 1-8 of 8 results.

A106454 Position of A014486(n-1) in A106455.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 16, 11, 14, 9, 12, 24, 21, 10, 15, 128, 64, 17, 32, 256, 65536, 13, 22, 29, 28, 56, 49, 18, 27, 192, 96, 45, 48, 384, 98304, 69, 42, 63, 20, 40, 107, 30, 255, 2048, 16384, 85, 512, 4096, 16777216, 273, 34, 51, 2097152, 1024, 257, 32768
Offset: 1

Views

Author

Antti Karttunen, May 09 2005

Keywords

Comments

See A106456.

Links

Crossrefs

Inverse: A106453. a(n) = A106442(A075164(n)). a(n) = A106452(n-1)+1. GF(2)[X]-analog of A075164.

A106453 Position of A106455(n) in A014486 plus one.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 12, 16, 10, 13, 24, 11, 17, 9, 20, 30, 66, 41, 15, 25, 68, 14, 198, 67, 31, 27, 26, 44, 627, 21, 2059, 53, 71, 83, 2057, 199, 73, 42, 6919, 39, 629, 69, 34, 204, 23715, 35, 29, 628, 54, 201, 200, 86, 82501, 28, 205, 72, 290513, 125, 1033413
Offset: 1

Views

Author

Antti Karttunen, May 09 2005

Keywords

Comments

See A106456.

Links

Crossrefs

Inverse: A106454. a(n) = A075163(A106443(n)). a(n) = A106451(n-1)+1. GF(2)[X]-analog of A075163.

A106451 Position of A106455(n+1) in A014486.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 4, 7, 11, 15, 9, 12, 23, 10, 16, 8, 19, 29, 65, 40, 14, 24, 67, 13, 197, 66, 30, 26, 25, 43, 626, 20, 2058, 52, 70, 82, 2056, 198, 72, 41, 6918, 38, 628, 68, 33, 203, 23714, 34, 28, 627, 53, 200, 199, 85, 82500, 27, 204, 71, 290512, 124, 1033412
Offset: 0

Views

Author

Antti Karttunen, May 09 2005

Keywords

Comments

See A106456.

Links

Crossrefs

Inverse: A106452. a(n) = A106453(n+1)-1. GF(2)[X]-analog of A075161.

A106452 Position of A014486(n) in A106455, minus one.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 4, 7, 15, 10, 13, 8, 11, 23, 20, 9, 14, 127, 63, 16, 31, 255, 65535, 12, 21, 28, 27, 55, 48, 17, 26, 191, 95, 44, 47, 383, 98303, 68, 41, 62, 19, 39, 106, 29, 254, 2047, 16383, 84, 511, 4095, 16777215, 272, 33, 50, 2097151, 1023, 256, 32767
Offset: 0

Views

Author

Antti Karttunen, May 09 2005

Keywords

Comments

See A106456. The next term, a(58) = 340282366920938463463374607431768211455 = (2^128) - 1 as A063171(58) = 1110101000, 11010100 = A063171(17), a(17)=127 and 127+1 = 128.

Links

Crossrefs

Inverse: A106451. a(n) = A106454(n+1)-1. GF(2)[X]-analog of A075162.

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.

A075165 Sequence A075166 interpreted as binary numbers and converted to decimal.

Original entry on oeis.org

0, 2, 10, 12, 42, 44, 170, 52, 50, 172, 682, 180, 2730, 684, 178, 56, 10922, 204, 43690, 692, 690, 2732, 174762, 184, 202, 10924, 210, 2740, 699050, 716, 2796202, 212, 2738, 43692, 714, 820, 11184810, 174764, 10930, 696, 44739242, 2764, 178956970
Offset: 1

Views

Author

Antti Karttunen, Sep 13 2002

Keywords

Crossrefs

Permutation of A014486. Same sequence shown in binary: A075166. The binary width of each term / 2 is given by A075167.

Formula

a(n) = A106455(A106442(n)). - Antti Karttunen, May 09 2005

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.
Showing 1-8 of 8 results.

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