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

Author

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

Keywords

• nonn
• more

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.

Examples

			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
...
		

Crossrefs

Cf. A355852, A030190, A210959, A027750.

Programs

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 *)