A275152 -id:A275152 - cp's OEIS frontend

A275157 Index when the partial tiling described in A275152 is perfect (i.e., all filled squares are contiguous).

1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 16, 19, 23, 25, 27, 28, 34, 35, 36, 37, 44, 48, 50, 53, 55, 60, 62, 63, 66, 67, 68, 69, 83, 87, 91, 93, 95, 100, 103, 106, 108, 110, 111, 113, 115, 120, 123, 126, 127, 130, 131, 132, 133, 134, 156, 165, 176, 181, 185

Offset: 1

Rémy Sigrist, Nov 13 2016

nonn

			The following table depicts the first partial tilings described in A275152 ("X" denotes a filled square); perfect tilings are marked as such:
Index  Perfect ?  Partial tiling described in A275152
-----  ---------  -----------------------------------
1      *          X
2      *          XX
3      *          XXXX
4      *          XXXXX
5                 XXXXXX X
6      *          XXXXXXXX
7      *          XXXXXXXXXX
8      *          XXXXXXXXXXXXX
9                 XXXXXXXXXXXXXX  X
10     *          XXXXXXXXXXXXXXXXXX
11                XXXXXXXXXXXXXXXXXXX X
12     *          XXXXXXXXXXXXXXXXXXXXXXX
13     *          XXXXXXXXXXXXXXXXXXXXXXXXX
14     *          XXXXXXXXXXXXXXXXXXXXXXXXXXXX
15     *          XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
16     *          XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
17                XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX   X
18                XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XX
19     *          XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
20                XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX X
21                XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX X
22                XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XX
23     *          XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
24                XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX X
25     *          XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

Rémy Sigrist, Table of n, a(n) for n = 1..2500

Cf. A275152.

A279125 Lexicographically earliest sequence such that, for any distinct i and j, a(i)=a(j) implies (i AND j)=0 (where AND stands for the bitwise AND operator).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Dec 06 2016

This sequence is similar to A279119 in the sense that here we check for common ones in binary representation and there we check for common prime factors.

By analogy with A275152, this sequence can be seen as a way to tile the first quadrant with fixed disconnected 2-dimensional polyominoes: the (vertical) polyomino corresponding to n is shifted to the right as little as possible so as not to overlap a previous polyomino, and a(n) gives the corresponding number of steps to the right (see illustration in Links section).

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms (by way of polyominos)
N. J. A. Sloane and Brady Haran, Amazing Graphs III, Numberphile video (2019).

Cf. A000079, A164346, A275152, A279119.

with(Bits):
n:= 100:
l:= []:
g:=[seq(0, i = 0..n-1)]:
for i from 1 to n by 1
do
a:= 0;
while (And(g[a + 1], i)) > 0
do
a++;
end do:
g[a + 1] += i;
l:= [op(l), a];
end do:
print(l); # Reza K Ghazi, Dec 29 2021

n = 100;
l = {};
g = ConstantArray[0, n];
For[i = 0, i < n, i++; a = 0; While[BitAnd[g[[a + 1]], i] > 0, a++];
  g[[a + 1]] += i;
  l = Append[l, a]];
l (* Reza K Ghazi, Dec 29 2021 *)

g = vector(72); for (n=1, #g, a = 0; while (bitand(g[a+1],n)>0, a++); g[a+1] += n; print1 (a", "))

n = 100
g = n * [0]
for i in range(1, n + 1):
    a = 0
    while g[a] & i:
        a += 1
    g[a] += i
    print(a, end=', ') # Reza K Ghazi, Dec 29 2021

a(n)=0 iff n belongs to A000079.

a(n)=1 iff n belongs to A164346.

A278388 Lexicographically earliest sequence such that (i2^a(i)) AND (j2^a(j)) = 0 for any distinct i and j (AND stands for the bitwise AND operator).

Original entry on oeis.org

0, 0, 2, 2, 5, 7, 10, 3, 13, 14, 18, 20, 24, 27, 31, 10, 35, 36, 41, 34, 44, 48, 53, 55, 60, 64, 69, 72, 77, 81, 86, 15, 51, 42, 61, 89, 93, 95, 101, 102, 108, 109, 115, 119, 123, 128, 134, 136, 138, 143, 145, 149, 155, 160, 166, 169, 175, 180, 186, 190, 196

Offset: 1

Rémy Sigrist, Nov 20 2016

By analogy with A275152, this sequence can be obtained by the following algorithm:
- we start with a half-open line of empty squares with coordinates X=0, X=1, X=2, etc.,
- for n=1, 2, 3, ...: we choose the least k such that the polyomino corresponding to n, shifted by k squares to the right, does not overlap one of the previous polyominoes.
a(2*k+1) > a(2*k) for any k>0.

			The following table depicts the first terms, alongside the corresponding polyominoes ("X" denotes a filled square, "_" denotes an empty square):
n     n in binary    a(n)    n as a polyomino shifted by a(n) to the right
--    -----------    ----    ---------------------------------------------
1     1              0       X
2     10             0       _X
3     11             2         XX
4     100            2         __X
5     101            5            X_X
6     110            7              _XX
7     111            10                XXX
8     1000           3          ___X
9     1001           13                   X__X
10    1010           14                    _X_X
11    1011           18                        XX_X
12    1100           20                          __XX
13    1101           24                              X_XX
14    1110           27                                 _XXX
15    1111           31                                     XXXX
16    10000          10                ____X
17    10001          35                                         X___X
18    10010          36                                          _X__X

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A275152.

sumn2a = 0; for (n=1, 1 000, a=0; while (bitand(sumn2a, n<

A286192 1-dimensional polyominoes (represented as integers) that do not tile a rectangle.

Original entry on oeis.org

27, 93, 99, 107, 111, 119, 123, 167, 183, 189, 215, 219, 223, 229, 231, 235, 237, 239, 247, 251

Offset: 1

Dmitry Kamenetsky, May 05 2017

The indices of the 1's in the binary representation of a number give the positions of the squares in the corresponding polyomino.

Erich Friedman, Tiling rectangles with one-dimensional disconnected polyominoes
Dmitry Kamenetsky and Tristrom Cooke, Tiling rectangles with holey polyominoes, arXiv:1411.2699 [cs.CG], 2015.

Cf. A235264, A275152.

cp's OEIS Frontend

A275157 Index when the partial tiling described in A275152 is perfect (i.e., all filled squares are contiguous).

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A279125 Lexicographically earliest sequence such that, for any distinct i and j, a(i)=a(j) implies (i AND j)=0 (where AND stands for the bitwise AND operator).

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A278388 Lexicographically earliest sequence such that (i2^a(i)) AND (j2^a(j)) = 0 for any distinct i and j (AND stands for the bitwise AND operator).

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Author

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Examples

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Programs

PARI

A286192 1-dimensional polyominoes (represented as integers) that do not tile a rectangle.

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Author

Keywords

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cp's OEIS Frontend

A275157 Index when the partial tiling described in A275152 is perfect (i.e., all filled squares are contiguous).

Views

Author

Keywords

Examples

Links

Crossrefs

A279125 Lexicographically earliest sequence such that, for any distinct i and j, a(i)=a(j) implies (i AND j)=0 (where AND stands for the bitwise AND operator).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A278388 Lexicographically earliest sequence such that (i*2^a(i)) AND (j*2^a(j)) = 0 for any distinct i and j (AND stands for the bitwise AND operator).

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Author

Keywords

Comments

Examples

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Programs

PARI

A286192 1-dimensional polyominoes (represented as integers) that do not tile a rectangle.

Views

Author

Keywords

Comments

Links

Crossrefs

A278388 Lexicographically earliest sequence such that (i2^a(i)) AND (j2^a(j)) = 0 for any distinct i and j (AND stands for the bitwise AND operator).