A355852 Numbers that can be written as the product of two of its divisors such that the binary value of the number has the same length as the concatenation of the binary values of the divisors and differs by only one bit from the divisor concatenation.

39, 78, 87, 156, 174, 183, 312, 348, 366, 375, 399, 539, 624, 696, 732, 750, 759, 798, 847, 1053, 1078, 1248, 1392, 1464, 1500, 1518, 1527, 1596, 1694, 1743, 2106, 2156, 2496, 2784, 2928, 3000, 3036, 3054, 3063, 3192, 3388, 3486, 3535, 3615, 4212, 4312, 4381, 4992, 5175, 5568, 5856, 6000

Offset: 1

Author

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

Keywords

• nonn

Comments

See A355857 for the smallest number that shares n out of n+1 bits with the divisor binary concatenation.

Examples

			39 is a term as 39 = 100111_2 = 13 * 3 = 1101_2 * 11_2, and "100111" has five bits out of six in common with "110111".
539 is a term as 539 = 1000011011_2 = 49 * 11 = 110001_2 * 1011_2 and "1000011011" has nine out of ten bits in common with "1100011011".
See the attached text file for other examples.
		

Links

• Michael De Vlieger, Plot of a(n) expanded in binary, black pixels indicating 1's and white 0's, with n on the x-axis and 2^y increasing from bottom to top.
• Scott R. Shannon, Divisor products for the first 1217 terms. These are all the terms up to 100000000.

Crossrefs

Cf. A355857, A030190, A210959, A027750.

Programs

Mathematica

Select[Range[6000], Function[{m, d}, 0 < Count[Map[Join @@ IntegerDigits[{##}, 2] & @@ {#, m/#} &, Divisors[m]], ?(Length[#] == Length[d] && Total[BitXor @@ {#, d}] == 1 &)]] @@ {#, IntegerDigits[#, 2]} &] (* _Michael De Vlieger, Jul 21 2022 *)