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 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 5 23:35 EDT 2024. Contains 374957 sequences. (Running on oeis4.)