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.
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
KEYWORD
nonn,more
AUTHOR
Scott R. Shannon and Michael De Vlieger, Jul 19 2022
STATUS
approved