

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.


4



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


LINKS



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



STATUS

approved



