A182622 -id:A182622 - 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

A182620 Triangle T(n,k) read by rows in which row n lists the divisors of n, written in base 2.

Original entry on oeis.org

1, 1, 10, 1, 11, 1, 10, 100, 1, 101, 1, 10, 11, 110, 1, 111, 1, 10, 100, 1000, 1, 11, 1001, 1, 10, 101, 1010, 1, 1011, 1, 10, 11, 100, 110, 1100, 1, 1101, 1, 10, 111, 1110, 1, 11, 101, 1111, 1, 10, 100, 1000, 10000, 1, 10001, 1, 10, 11

Offset: 1

Omar E. Pol, Nov 22 2010

Numbers of triangle A027750, written in base 2.

			The divisors of 10 are 1, 2, 5, 10 then row 10 lists the binary numbers 1, 10, 101, 1010.
Triangle begins:
1,
1, 10,
1, 11,
1, 10, 100,
1, 101,
1, 10, 11, 110,
1, 111,
1, 10, 100, 1000,
1, 11, 1001,
1, 10, 101, 1010,
1, 1011,
1, 10, 11, 100, 110, 1100,

Nathaniel Johnston, Table of n, a(n) for n = 1..7069

Cf. A007088, A027750, A182621, A182622, A182623, A182624, A182630.

with(numtheory):for n from 1 to 10 do for d in divisors(n) do printf("%d, ",convert(d,binary)); od:printf("\n");od: # Nathaniel Johnston, Apr 19 2011

Table[FromDigits[IntegerDigits[#,2]]&/@Divisors[n],{n,20}]//Flatten (* Harvey P. Dale, May 31 2018 *)

T(n,k) = A007088(A027750(n,k)).

a(38)-a(55) from Nathaniel Johnston, Apr 19 2011

A182624 Primes in A182623.

Original entry on oeis.org

7, 13, 29, 61, 101, 107, 199, 211, 229, 241, 419, 449, 467, 479, 769, 823, 829, 859, 991, 1009, 1021, 1571, 1601, 1637, 1667, 1697, 1733, 1811, 1847, 1877, 1901, 1907, 1931, 3079, 3109, 3229, 3271, 3307, 3331, 3457, 3499, 3529, 3541, 3547

Offset: 1

Omar E. Pol, Nov 23 2010

Nathaniel Johnston, Table of n, a(n) for n = 1..2000

Cf. A000040, A182620, A182621, A182622, A182623.

lim:=1800: with(numtheory):A182624:={}:for n from 1 to lim do s:="": for d in divisors(n) do s:= cat(s,convert(convert(d, binary),string)): od: m:=convert(parse(s),decimal,binary):if(isprime(m))then A182624:=A182624 union {m};fi: od:
A182624:=sort(convert(A182624,list)):for n from 1 to nops(A182624) do if(A182624[n]>2*lim)then break:fi:printf("%d, ",A182624[n]):od: # Nathaniel Johnston, Apr 19 2011

More terms from Vincenzo Librandi, Jan 30 2011

a(16) - a(44) from Nathaniel Johnston, Apr 19 2011

A182623 Numbers n with property that there is a number m such that if we concatenate the binary representations of the divisors of m in increasing order we get the binary representation of n.

Original entry on oeis.org

1, 6, 7, 13, 15, 27, 29, 49, 51, 52, 55, 61, 63, 101, 105, 107, 111, 117, 121, 123, 125, 195, 199, 201, 207, 211, 217, 222, 225, 229, 231, 235, 237, 241, 255, 387, 393, 395, 405, 407, 413, 419, 423, 429, 435, 437, 441, 447, 449, 453, 455, 467, 479, 483, 485, 489, 495, 497, 507, 769, 775, 781, 783, 789

Offset: 1

Omar E. Pol, Nov 23 2010

Numbers of A182622 in increasing order (without repetitions).

			858 written in base 2 is 1101011010. The string "1101011010" may be broken up into four parts (1, 10, 101, 1010) that are also the binary representations of a sequence of numbers in increasing order: 1, 2, 5, 10. These numbers are the divisors of 10. Then 858 is in the sequence.

Cf. A007088, A027750, A182620, A182621, A182622, A182624.

A182628 Triangle T(n,k) read by rows in which row n lists the number of digits of the binary expansion of the divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 3, 1, 3, 1, 2, 2, 3, 1, 3, 1, 2, 3, 4, 1, 2, 4, 1, 2, 3, 4, 1, 4, 1, 2, 2, 3, 3, 4, 1, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 5, 1, 2, 2, 3, 4, 5, 1, 5, 1, 2, 3, 3, 4, 5, 1, 2, 3, 5, 1, 2, 4, 5, 1, 5, 1, 2, 2

Offset: 1

Omar E. Pol, Nov 23 2010

Row n lists the number of digits of the numbers in the n-th row of triangle A182620.

			Triangle begins:
1,
1, 2,
1, 2,
1, 2, 3,
1, 3,
1, 2, 2, 3,
1, 3,
1, 2, 3, 4,
1, 2, 4,
1, 2, 3, 4,
1, 4,
1, 2, 2, 3, 3, 4,

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A182620, A182621, A182622.

Row sums give A182627.

with(numtheory):for n from 1 to 12 do for d in divisors(n) do printf("%d, ",length(convert(convert(d, binary),string))); od:printf("\n"); od: # Nathaniel Johnston, Apr 19 2011

a(38)-a(79) from Nathaniel Johnston, Apr 19 2011

A231116 Numbers k such that the total number of digits of all the divisors of k written in base 2 is equal to k.

Original entry on oeis.org

1, 3, 10, 24

Offset: 1

Jaroslav Krizek, Nov 03 2013

Sequence is finite with 4 terms.
Numbers k such that the concatenation of their divisors written in base 2 contains k digits.
Subsequence of finite sequence:  1, 2, 3, 4, 6, 8, 10, 12, 24 (numbers k such that the total number of digits of all the divisors of k written in base 2 is greater than or equal to k).
Numbers k such that A182627(k) = k.

			24 is in the sequence because the concatenation of the divisors of 24 in base 2 [(1, 10, 11, 100, 110, 1000, 1100, 11000) -> 110111001101000110011000] contains 24 digits.

Cf. A182622, A182627.

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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A182620 Triangle T(n,k) read by rows in which row n lists the divisors of n, written in base 2.

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A182624 Primes in A182623.

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A182623 Numbers n with property that there is a number m such that if we concatenate the binary representations of the divisors of m in increasing order we get the binary representation of n.

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A182628 Triangle T(n,k) read by rows in which row n lists the number of digits of the binary expansion of the divisors of n.

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A231116 Numbers k such that the total number of digits of all the divisors of k written in base 2 is equal to k.

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