cp's OEIS Frontend

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.

%I A355857 #15 Jul 26 2022 12:51:20
%S A355857 39,78,156,312,539,1053,2106,4212,8299,16598,32889,65778,131499,
%T A355857 262605,525210,1049073,2098146,4196292,8392584,16785168,33556449,
%U A355857 67112898,134225465,268450930
%N 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.
%C A355857 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.
%C A355857 See A355852 for further details.
%e A355857 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".
%e A355857 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".
%e A355857 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".
%e A355857 Table showing a(n) in binary, replacing 0's with "." to accentuate the pattern of 1's:
%e A355857                          Binary     Decimal
%e A355857                          1..111 =        39
%e A355857                         1..111. =        78
%e A355857                        1..111.. =       156
%e A355857                       1..111... =       312
%e A355857                      1....11.11 =       539
%e A355857                     1.....111.1 =      1053
%e A355857                    1.....111.1. =      2106
%e A355857                   1.....111.1.. =      4212
%e A355857                  1......11.1.11 =      8299
%e A355857                 1......11.1.11. =     16598
%e A355857                1........1111..1 =     32889
%e A355857               1........1111..1. =     65778
%e A355857              1........11.1.1.11 =    131499
%e A355857             1.........111..11.1 =    262605
%e A355857            1.........111..11.1. =    525210
%e A355857           1...........11111...1 =   1049073
%e A355857          1...........11111...1. =   2098146
%e A355857         1...........11111...1.. =   4196292
%e A355857        1...........11111...1... =   8392584
%e A355857       1...........11111...1.... =  16785168
%e A355857      1..............111111....1 =  33556449
%e A355857     1..............111111....1. =  67112898
%e A355857    1..............1111...111..1 = 134225465
%e A355857   1..............1111...111..1. = 268450930
%e A355857 ...
%t A355857 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 *)
%Y A355857 Cf. A355852, A030190, A210959, A027750.
%K A355857 nonn,more
%O A355857 5,1
%A A355857 _Scott R. Shannon_ and _Michael De Vlieger_, Jul 19 2022