A355852 -id:A355852 - 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.

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

Offset: 5

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

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.

			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
...

Cf. A355852, A030190, A210959, A027750.

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 *)

A355790 Numbers that can be written as the product of two divisors greater than 1 such that the number is contained in the string concatenation of the divisors.

Original entry on oeis.org

64, 95, 110, 210, 325, 510, 624, 640, 664, 950, 995, 1010, 1100, 1110, 3250, 3325, 5134, 6240, 6400, 6640, 6664, 7125, 7616, 8145, 9500, 9950, 9995, 11000, 11100, 11110, 20100, 21052, 21175, 25100, 26208, 32500, 33250, 33325, 35126, 50100, 51020, 51204, 51340, 57125, 62400, 64000, 65114

Offset: 1

Scott R. Shannon, Jul 17 2022

			64 is a term as 64 = 16 * 4 and "16" + "4" = "164" contains "64".
65114 is a term as 65114 = 4651 * 14 and "4651" + "14" = "465114" contains "65114".
See the attached text file for other examples.

Scott R. Shannon, Divisor product of the first 232 terms. These are all the numbers up to 50000000.

Cf. A355791 (base-2), A030190, A210959, A027750, A355852, A339144, A341035, A342127, A027748, A048991, A330027, A077293.

from sympy import divisors
def ok(n):
    s, divs = str(n), divisors(n)[1:-1]
    return any(s in str(d)+str(n//d) for d in divs)
print([k for k in range(1, 10**5) if ok(k)]) # Michael S. Branicky, Jul 27 2022

A355791 Numbers that can be written as the product of two divisors greater than 1 such that the number in binary is contained in the string concatenation of the divisors in binary.

Original entry on oeis.org

6, 10, 12, 14, 24, 28, 30, 36, 42, 48, 56, 57, 60, 62, 96, 112, 120, 124, 126, 136, 170, 192, 224, 240, 248, 252, 254, 292, 355, 384, 448, 480, 496, 504, 508, 510, 528, 682, 737, 768, 896, 921, 960, 992, 1008, 1016, 1020, 1022, 1536, 1792, 1920, 1984, 2016, 2032, 2040, 2044, 2046, 2080, 2184, 2340

Offset: 1

Scott R. Shannon, Jul 17 2022

			6 is a term as 6 = 110_2 = 3 * 2 = 11_2 * 10_2 and "11" + "10" = "1110" contains "110".
2340 is a term as 2340 = 100100100100_2 = 4 * 585 = 100_2 * 1001001001_2 and "100" + "1001001001" contains "100100100100".
See the attached text file for other examples.

Scott R. Shannon, Divisor product of the first 417 terms. These are all the numbers up to 100000000.

Cf. A355790 (base-10), A355852, A355857, A030190, A355852, A210959, A027750.

q[n_] := AnyTrue[Rest @ Most @ Divisors[n], StringContainsQ[StringJoin @@ IntegerString[{#, n/#}, 2], IntegerString[n, 2]] &]; Select[Range[2, 2500], q] (* Amiram Eldar, Jul 27 2022 *)

from sympy import divisors
def ok(n):
    b, divs = bin(n)[2:], divisors(n)[1:-1]
    return any(b in bin(d)[2:]+bin(n//d)[2:] for d in divs)
print([k for k in range(1, 2400) if ok(k)]) # Michael S. Branicky, Jul 27 2022

A356143 Numbers that can be written in two or more ways as the product of three divisors greater than 1 such that the number in binary is contained in the string concatenation of the divisors in binary.

Original entry on oeis.org

126, 252, 495, 504, 1008, 2016, 3420, 3510, 4032, 5850, 8064, 11700, 16128, 23400, 32256, 46800, 64512, 93600, 129024, 187200, 258048, 374400, 516096, 748800, 1032192, 1497600, 2064384, 2995200, 4128768, 5990400, 8257536, 11980800, 16515072, 23961600, 33030144, 47923200, 57587274, 66060288

Offset: 1

Scott R. Shannon, Jul 30 2022

There are thirty-nine terms less than 100 million - the terms in the data section along with 95846400. All of these numbers have two ways that they can be written to satisfy their binary value being contained in the binary concatenation of the three divisors; it is unknown if numbers exist that can be written in three or more ways that satisfy this criterion. The only odd number in the first thirty-nine terms is 495; it is unknown if more exist.

			126 is a term as 126 = 1111110_2 = 3 * 3 * 14 = 11_2 * 11_2 * 1110_2 = 3 * 7 * 6 = 11_2 * 111_2 * 110_2 and "11" + "11" + "1110" = "11111110" contains "1111110" and "11" + "111" + "110" = "11111110" contains "1111110".
3420 is a term as 3420 = 110101011100_2 = 5 * 171 * 4 = 101_2 * 10101011_2 * 100_2 = 6 * 10 * 57 = 110_2 * 1010_2 * 111001_2 and "101" + "10101011" + "100" = "10110101011100" contains "110101011100" and "110" + "1010" + "111001" = "1101010111001" contains "110101011100".
See the attached text file for other examples.

Scott R. Shannon, Divisor products of the first thirty-nine terms. These are all the numbers up to 100 million.

Cf. A356176, A355791, A355790, A355852, A355857, A030190, A355852, A210959, A027750.

A356176 Numbers that can be written as the product of three divisors greater than 1 such that the number in binary is contained in the string concatenation of the divisors in binary.

Original entry on oeis.org

30, 60, 63, 90, 110, 120, 126, 180, 220, 228, 240, 252, 340, 350, 430, 440, 480, 495, 504, 606, 612, 680, 759, 840, 860, 880, 891, 927, 957, 960, 990, 1008, 1212, 1224, 1320, 1526, 1710, 1720, 1755, 1760, 1782, 1854, 1920, 1980, 2016, 2376, 2424, 2466, 2640, 2925, 2955, 2990, 3045, 3050

Offset: 1

Scott R. Shannon, Jul 28 2022

See A356143 for the list of numbers that have two or more ways to be written as the product of three divisors such that the number in binary is contained in the string concatenation of the divisors in binary.

			30 is a term as 30 = 11110_2 = 5 * 3 * 2 = 101_2 * 11_2 * 10_2 and "101" + "11" + "10" = "1011110" contains "11110".
3050 is a term as 3050 = 101111101010_2 = 5 * 122 * 5 = 101_2 * 1111010_2 * 101_2 and "101" + "1111010" + "101" = "1011111010101" contains "101111101010".
See the attached text file for other examples.

Scott R. Shannon, Divisor products up to 100 million.

Cf. A356143, A355791, A355790, A355852, A355857, A030190, A355852, A210959, A027750.

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A355790 Numbers that can be written as the product of two divisors greater than 1 such that the number is contained in the string concatenation of the divisors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

A355791 Numbers that can be written as the product of two divisors greater than 1 such that the number in binary is contained in the string concatenation of the divisors in binary.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

A356143 Numbers that can be written in two or more ways as the product of three divisors greater than 1 such that the number in binary is contained in the string concatenation of the divisors in binary.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A356176 Numbers that can be written as the product of three divisors greater than 1 such that the number in binary is contained in the string concatenation of the divisors in binary.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs