A182621 -id:A182621 - cp's OEIS frontend

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

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

A182622 a(n) is the number whose binary representation is the concatenation of the divisors of n written in base 2.

Original entry on oeis.org

1, 6, 7, 52, 13, 222, 15, 840, 121, 858, 27, 28268, 29, 894, 991, 26896, 49, 113970, 51, 215892, 2037, 3446, 55, 14471576, 441, 3514, 3899, 217052, 61, 14538238, 63, 1721376, 7905, 13410, 7139, 926213284, 101, 13542, 8039, 221009192

Offset: 1

Omar E. Pol, Nov 22 2010

a(n) is A182621(n), interpreted as a binary number, written in base 10. The first repeated element is 991, from 15 and 479.

Except for 1, no power of 2 can occur in this sequence, an obvious consequence of the fact that a(n) has to be the sum of at least two distinct powers of 2 for all n > 1. - Alonso del Arte, Nov 13 2013

			The divisors of 10 are 1, 2, 5, 10. Then 1, 2, 5, 10 written in base 2 are 1, 10, 101, 1010. The concatenation of 1, 10, 101, 1010 is 1101011010. Then a(10) = 858 because the binary number 1101011010 written in base 10 is 858.

Indranil Ghosh, Table of n, a(n) for n = 1..50000

Cf. A027750, A007088, A182620, A182621, A182623, A182624, A182627, A182632.

concatBits[n_] := FromDigits[Join @@ (IntegerDigits[#, 2]& /@ Divisors[n]), 2]; concatBits /@ Range[40](* Giovanni Resta, Nov 23 2010 *)

a(n) = {my(cbd = []); fordiv(n, d, cbd = concat(cbd, binary(d));); fromdigits(cbd, 2);} \\ Michel Marcus, Jan 28 2017

def A182622(n):
    s=""
    for i in range(1,n+1):
        if n%i==0:
            s+=bin(i)[2:]
    return int(s,2) # Indranil Ghosh, Jan 28 2017

a(p) = 2^(floor(log_2(p)) + 1) + p for p prime. Also, a(p + k) > a(p) for all k > 0. Furthermore, for all primes p > 3, a(p) < a(p - 1).

a(2^(m - 1)) = sum(k = 0 .. m - 1, 2^((m^2 + m)/2 - (k^2 + k)/2 - 1)) = A164894(m). - Alonso del Arte, Nov 13 2013

More terms from Giovanni Resta, Nov 23 2010

A182627 Total number of digits in binary expansion of all divisors of n.

Original entry on oeis.org

1, 3, 3, 6, 4, 8, 4, 10, 7, 10, 5, 15, 5, 10, 10, 15, 6, 17, 6, 18, 11, 12, 6, 24, 9, 12, 12, 18, 6, 24, 6, 21, 13, 14, 13, 30, 7, 14, 13, 28, 7, 26, 7, 21, 20, 14, 7, 35, 10, 21, 14, 21, 7, 28, 14, 28, 14, 14, 7, 42, 7, 14, 21, 28, 15, 30, 8, 24, 15, 30, 8

Offset: 1

Omar E. Pol, Nov 23 2010

Also, total number of digits in row n of triangle A182620.
Also, number of digits of A182621(n).
Rows sums of triangle A182628.
From Davide Rotondo, Apr 20 2022: (Start)
Can be constructed by writing the sequence of natural numbers with 1 one, 2 twos, 4 threes, 8 fours, ..., where 1,2,4,8,... are consecutive powers of 2; then the same sequence spaced by a zero, then the same sequence spaced by two zeros, and so on. Finally add the values of the columns.
     1  2  2  3  3  3  3  4  4  4  4  4  4  4  4  5 ...
     0  1  0  2  0  2  0  3  0  3  0  3  0  3  0  4 ...
     0  0  1  0  0  2  0  0  2  0  0  3  0  0  3  0 ...
     0  0  0  1  0  0  0  2  0  0  0  2  0  0  0  3 ...
     0  0  0  0  1  0  0  0  0  2  0  0  0  0  2  0 ...
     0  0  0  0  0  1  0  0  0  0  0  2  0  0  0  0 ...
     0  0  0  0  0  0  1  0  0  0  0  0  0  2  0  0 ...
     0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  2 ...
     0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0 ...
     0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0 ...
     0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0 ...
     0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0 ...
     0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0 ...
     0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0 ...
     0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0 ...
     0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1 ...
     ...
     ----------------------------------------------
Tot. 1  3  3  6  4  8  4 10  7 10  5 15  5 10 10 15 ...  (End)

			The divisors of 12 are 1, 2, 3, 4, 6, 12. These divisors written in base 2 are 1, 10, 11, 100, 110, 1100. Then a(12)=15 because 1+2+2+3+3+4 = 15.

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

Cf. A093653, A135539, A182620, A182621, A182628.
Cf. A093653 (number of 1's in binary expansion of all divisors of n).
Cf. A226590 (number of 0's in binary expansion of all divisors of n).

Table[Total[IntegerLength[Divisors[n],2]],{n,60}] (* Harvey P. Dale, Jan 26 2012 *)

a(n) = sumdiv(n, d, 1+logint(d, 2)); \\ Michel Marcus, Dec 11 2020

from sympy import divisors
def a(n): return sum(d.bit_length() for d in divisors(n))
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Apr 21 2022

a(n) = A093653(n) + A226590(n). - Jaroslav Krizek, Sep 01 2013
a(n) = tau(n) + Sum_{d|n} floor(log_2(d)). - Ridouane Oudra, Dec 11 2020
a(n) = Sum_{i=0..floor(log_2(n))} A135539(n,2^i). - Ridouane Oudra, Sep 19 2022

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.

A226590 Total number of 0's in binary expansion of all divisors of n.

Original entry on oeis.org

0, 1, 0, 3, 1, 2, 0, 6, 2, 4, 1, 6, 1, 2, 1, 10, 3, 7, 2, 9, 2, 4, 1, 12, 3, 4, 3, 6, 1, 6, 0, 15, 5, 8, 4, 15, 3, 6, 3, 16, 3, 8, 2, 9, 5, 4, 1, 20, 3, 9, 5, 9, 2, 10, 3, 12, 4, 4, 1, 15, 1, 2, 4, 21, 7, 14, 4, 15, 5, 12, 3, 26, 4, 8, 6, 12, 4, 10, 2, 25, 7

Offset: 1

Jaroslav Krizek, Aug 31 2013

Also total number of 0's in binary expansion of concatenation of the binary numbers that are the divisors of n written in base 2 (A182621).

a(n) = 0 iff n = 1 or n is a Mersenne prime (A000668). - Bernard Schott, Apr 22 2022

			a(8) = 6 because the divisors of 8 are [1, 2, 4, 8] and in binary: 1, 10, 100, 1000, so six 0's.

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

Cf. A093653 (number of 1's in binary expansion of all divisors of n).
Cf. A182627 (number of digits in binary expansion of all divisors of n).
Cf. A182621 (concatenation of the divisors of n written in base 2).
Cf. A000217, A000668.

Table[Count[Flatten[IntegerDigits[Divisors[n], 2]], 0], {n, 81}] (* T. D. Noe, Sep 04 2013 *)

a(n) = sumdiv(n, d, 1+logint(d, 2) - hammingweight(d)); \\ Michel Marcus, Apr 24 2022

from sympy import divisors
def a(n): return sum(bin(d)[2:].count("0") for d in divisors(n))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Apr 20 2022

a(n) = A182627(n) - A093653(n).

a(2^n) = n*(n+1)/2 = A000217(n). - Bernard Schott, Apr 22 2022

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

A229086 Numbers k such that the concatenation of the divisors of k written in base 2 is a palindromic number.

Original entry on oeis.org

1, 3, 7, 11, 19, 31, 43, 67, 103, 127, 131, 199, 239, 307, 331, 379, 439, 463, 547, 683, 887, 911, 991, 1123, 1171, 1291, 1531, 1543, 1783, 1951, 2731, 2843, 3067, 3079, 3511, 3823, 4099, 5107, 5323, 5419, 5659, 5851, 6151, 6343, 6679, 6871, 6967, 7159, 8191

Offset: 1

Jaroslav Krizek, Sep 13 2013

Concatenation of the divisors of n written in base 2 = A182621.
Are there composite numbers with this property?
Conjecture: a(n) for n > 1 is not equal to A161387, sequence also contains composites.
The answer to the question above is Yes. The first two composites are 297883 and 476635. - T. D. Noe, Sep 17 2013

			The number 103 is in sequence because the concatenation of its divisors written in base 2 (1, 1100111) is palindromic number - 11100111.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A161387, A182621.

Select[Range[10000], (temp=Flatten[IntegerDigits[Divisors[#],2]]; temp==Reverse[temp])&]

cp's OEIS Frontend

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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A182622 a(n) is the number whose binary representation is the concatenation of the divisors of n written in base 2.

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A182627 Total number of digits in binary expansion of all divisors of n.

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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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A226590 Total number of 0's in binary expansion of all divisors 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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