A182631 -id:A182631 - cp's OEIS frontend

A182621 a(n) is the concatenation of the binary numbers that are the divisors of n written in base 2.

1, 110, 111, 110100, 1101, 11011110, 1111, 1101001000, 1111001, 1101011010, 11011, 110111001101100, 11101, 1101111110, 1111011111, 110100100010000, 110001, 11011110100110010, 110011, 110100101101010100, 11111110101, 110101110110, 110111, 110111001101000110011000

Offset: 1

Omar E. Pol, Nov 22 2010

a(n) is the concatenation of the numbers of row n of triangle A182620. The first repeated element is a(15) = 1111011111 = a(479), where a(15) is the concatenation of 1, 11, 101 and 1111 but a(479) is the concatenation of 1 and 111011111. See A182620 and A182622 for more information.

			The divisors of 10 are 1, 2, 5, 10, which written in base 2 are 1, 10, 101, 1010. The concatenation of 1, 10, 101, 1010 is 1101011010, so a(10) = 1101011010.

Jaroslav Krizek, Table of n, a(n) for n = 1..500

Cf. A007088, A027750, A182620, A182622, A182623, A182624, A182631.

A182621[n_]:=FromDigits[Flatten[IntegerDigits[Divisors[n],2]]];Array[A182621,50] (* Paolo Xausa, Aug 31 2023 *)

a(n) = fromdigits(concat(apply(binary,divisors(n)))); \\ Kevin Ryde, May 02 2023

from sympy import divisors
def a(n): return int("".join(bin(d)[2:] for d in divisors(n)))
print([a(n) for n in range(1, 25)]) # Michael S. Branicky, Apr 20 2022

A182621 = lambda n: Integer(''.join(d.str(base=2) for d in divisors(n))) # D. S. McNeil, Dec 19 2010

A182630 T(n,k) = A002024(k+1)*n + A002262(k), n >= 0, k >= 0, read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 2, 2, 1, 3, 4, 3, 0, 4, 6, 5, 3, 1, 5, 8, 7, 6, 4, 2, 6, 10, 9, 9, 7, 5, 0, 7, 12, 11, 12, 10, 8, 4, 1, 8, 14, 13, 15, 13, 11, 8, 5, 2, 9, 16, 15, 18, 16, 14, 12, 9, 6, 3, 10, 18, 17, 21, 19, 17, 16, 13, 10, 7, 0

Offset: 0

Omar E. Pol, Dec 06 2010

A table of congruences.

See A182631 for another version.

			Table of congruences:
===============+====+=======+==========+=============+====
           mod |  1 |   2   |     3    |      4      | ...
===============+====+=======+==========+=============+====
  congruent to |  0 |  0  1 |  0  1  2 |  0  1  2  3 | ...
===============+====+=======+==========+=============+====
Array begins:  |  0 |  0  1 |  0  1  2 |  0  1  2  3 | ...
               |  1 |  2  3 |  3  4  5 |  4  5  6  7 | ...
               |  2 |  4  5 |  6  7  8 |  8  9 10 11 | ...
               |  3 |  6  7 |  9 10 11 | 12 13 14 15 | ...
               |  4 |  8  9 | 12 13 14 | 16 17 18 19 | ...
               |  5 | 10 11 | 15 16 17 | 20 21 22 23 | ...
               |  6 | 12 13 | 18 19 20 | 24 25 26 27 | ...
               |  7 | 14 15 | 21 22 23 | 28 29 30 31 | ...
               |  8 | 16 17 | 24 25 26 | 32 33 34 35 | ...
               |  9 | 18 19 | 27 28 29 | 36 37 38 39 | ...
               | 10 | 20 21 | 30 31 32 | 40 41 42 43 | ...

Columns: A001477, A005483, A005408, A008585, A016777, A016789, A008586, A016813, A016825, A004767.
Rows: A002262, A094727, A182815.
Cf. A182631.

A002262 := proc(n) n-binomial(floor(1/2+sqrt(2+2*n)),2) ; end proc:
A002024 := proc(n) ceil((sqrt(1+8*n)-1)/2) ;end proc:
A182630 := proc(n,k) A002024(k+1)*n+A002262(k) ; end proc: # R. J. Mathar, Dec 09 2010

A182814 Main diagonal of table A182630.

Original entry on oeis.org

0, 2, 5, 9, 13, 17, 24, 29, 34, 39, 50, 56, 62, 68, 74, 90, 97, 104, 111, 118, 125, 147, 155, 163, 171, 179, 187, 195, 224, 233, 242, 251, 260, 269, 278, 287, 324, 334, 344, 354, 364, 374, 384, 394, 404, 450, 461, 472, 483, 494, 505, 516, 527, 538, 549, 605, 617, 629, 641, 653, 665, 677, 689, 701, 713, 725, 792, 805, 818, 831, 844, 857, 870, 883, 896, 909, 922, 935, 1014

Offset: 0

Omar E. Pol, Dec 06 2010

Main diagonal of a table of congruences.

Cf. A182630, A182631.

A182814 := proc(n) A182630(n,n) ; end proc:
seq(A182814(n),n=0..80) ;

a(n) = A182630(n,n).

A182810 Array read by antidiagonals which lists the partition number of the numbers of the table A182630.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 5, 11, 7, 3, 1, 7, 22, 15, 11, 5, 2, 11, 42, 30, 30, 15, 7, 1, 15, 77, 56, 77, 42, 22, 5, 1, 22, 135, 101, 176, 101, 56, 22, 7, 2, 30, 231, 176, 385, 231, 135, 77, 30, 11, 3, 42, 385, 297, 792, 490, 297, 231, 101, 42, 15, 1

Offset: 0

Omar E. Pol, Dec 06 2010

			...1....1....1....1....1....2....1....1....2....3....1.
...1....2....3....3....5....7....5....7...11...15....7.
...2....5....7...11...15...22...22...30...42...56...42.
...3...11...15...30...42...56...77..101..135..176..176.
...5...22...30...77..101..135..231..297..385..490..627.
...7...42...56..176..231..297..627..792.1002.1255.1958.

Cf. A000041, A182630, A182631, A182814.

A182810 := proc(n,k) combinat[numbpart](A182630(n,k)) ; end proc:

T(n,k) = A000041(A182630(n,k)), n,k >=0.

A182815 The third row of table A182630.

Original entry on oeis.org

2, 4, 5, 6, 7, 8, 8, 9, 10, 11, 10, 11, 12, 13, 14, 12, 13, 14, 15, 16, 17, 14, 15, 16, 17, 18, 19, 20, 16, 17, 18, 19, 20, 21, 22, 23, 18, 19, 20, 21, 22, 23, 24, 25, 26, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 26

Offset: 0

Omar E. Pol, Dec 06 2010

Cf. A002262, A094727, A182630, A182631.

Maple
```
seq(A182630(2,n),n=0..80) ;
```

a(n) = A182630(2,n).

A182692 Composite Beatty sequence of sqrt(3).

Original entry on oeis.org

2, 3, 7, 12, 28, 48, 113, 195, 461, 798, 1888, 3270, 7736, 13399, 31702, 54909, 129916, 225021, 532405, 922152, 2181835, 3779049, 8941325, 15486829, 36642230, 63466204, 150162650, 260089339, 615377983, 1065865932

Offset: 1

Clark Kimberling, Nov 27 2010

nonn

The bisection (3,12,48,...) is a subsequence of A022838.
The bisection (2,7,28,...) is a subsequence of A054406.
See the comment at A107857 regarding Beatty sequences.

Cf. A022838, A054406, A107857, A182631.

a(n)=floor(s*a(n-1)) if n odd, a(n)=floor(r*a(n-1)) if n even, where r=sqrt(3), s=(r+3)/2, a(1)=floor(s).

cp's OEIS Frontend

A182621 a(n) is the concatenation of the binary numbers that are the divisors of n written in base 2.

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A182630 T(n,k) = A002024(k+1)*n + A002262(k), n >= 0, k >= 0, read by antidiagonals.

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A182814 Main diagonal of table A182630.

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A182810 Array read by antidiagonals which lists the partition number of the numbers of the table A182630.

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A182815 The third row of table A182630.

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A182692 Composite Beatty sequence of sqrt(3).

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