A355857 The smallest number in A355852 whose binary value shares n out of n+1 bits with the concatenation of the binary values of its divisors' product.

39, 78, 156, 312, 539, 1053, 2106, 4212, 8299, 16598, 32889, 65778, 131499, 262605, 525210, 1049073, 2098146, 4196292, 8392584, 16785168, 33556449, 67112898, 134225465, 268450930

Offset

5,1

Comments

The sequence starts at n = 5 as there are no numbers whose binary value shares n out of n+1 bits with their binary divisor concatenations for n <= 4. In general each term is twice or very close to twice the previous term, although this does not hold true for a(4) to a(5), implying other terms which are significantly lower than twice the previous term may also exist.

See A355852 for further details.

Links

Table of n, a(n) for n=5..28.

Example

a(5) = 39 as 39 = 100111_2 = 13 * 3 = 1101_2 * 11_2, and "100111" has five bits out of six in common with "110111".
a(7) = 156 as 156 = 10011100_2 = 13 * 12 = 1101_2 * 1100_2, and "10011100" has seven bits out of eight in common with "11011100".
a(10) = 1053 as 1053 = 10000011101_2 = 81 * 13 = 1010001_2 * 1101_2, and "10000011101" has ten bits out of eleven in common with "10100011101".
Table showing a(n) in binary, replacing 0's with "." to accentuate the pattern of 1's:
                         Binary     Decimal
                         1..111 =        39
                        1..111. =        78
                       1..111.. =       156
                      1..111... =       312
                     1....11.11 =       539
                    1.....111.1 =      1053
                   1.....111.1. =      2106
                  1.....111.1.. =      4212
                 1......11.1.11 =      8299
                1......11.1.11. =     16598
               1........1111..1 =     32889
              1........1111..1. =     65778
             1........11.1.1.11 =    131499
            1.........111..11.1 =    262605
           1.........111..11.1. =    525210
          1...........11111...1 =   1049073
         1...........11111...1. =   2098146
        1...........11111...1.. =   4196292
       1...........11111...1... =   8392584
      1...........11111...1.... =  16785168
     1..............111111....1 =  33556449
    1..............111111....1. =  67112898
   1..............1111...111..1 = 134225465
  1..............1111...111..1. = 268450930
...

Mathematica

s = Select[Range[2^16], Function[{m, d}, 0 < Count[Map[Join @@ IntegerDigits[{##}, 2] & @@ {#, m/#} &, Divisors[m]], _?(Length[#] == Length[d] && Total[BitXor @@ {#, d}] == 1 &)]] @@ {#, IntegerDigits[#, 2]} &]; t = Array[Function[{m, d, k}, Length[#] - FirstPosition[#, 1][[1]] & /@ Select[Map[BitXor @@ {#, d} &, Select[Map[Apply[Join, IntegerDigits[{#, m/#}, 2]] &, Most@ Rest@ Divisors[#1]], Length[#] == k &]], Total@ # == 1 &]] @@ {#1, #2, Length[#2]} & @@ {#, IntegerDigits[#, 2]} &@ s[[#]] &, Length[s]][[All, 1]]; Map[s[[FirstPosition[t, #][[1]]]] &, Union@ FoldList[Max, t]] (* Michael De Vlieger, Jul 22 2022 *)

Crossrefs

Cf. A355852, A030190, A210959, A027750.

Sequence in context: A072122 A355852 A290815 * A354227 A063335 A020264

Adjacent sequences: A355854 A355855 A355856 * A355858 A355859 A355860

Keyword

nonn,more

Author

Scott R. Shannon and Michael De Vlieger, Jul 19 2022

Status

approved