A346219 Base-10 numbers k such that k can be written as k = A * B and R(k) = R(A) * R(B) in six or more bases, from base 2 to base 10, and where R(k), the digit reversal of k, is read as a number in the same base.

1122, 17875, 65331, 367598, 818545, 1997905, 43998955, 100383283, 112887775, 112977865, 145683265, 230034805, 5231187650

Offset: 1

Author

Scott R. Shannon, Eric Angelini and Carole Dubois, Jul 11 2021

Keywords

• nonn
• base
• more

Comments

This is a variation of the sequence A346133. Similar rules are used to determine the allowed values of A and B - neither number can have a leading 0, and both cannot be palindromes. However the reverse of k may appear as in general any solutions for k and R(k) will occur in different bases.

This sequence lists those base-10 numbers that meet these criteria in six or more bases, from base 2 to base 10. Note that, although k must stay the same when written in the different bases, the values of A and B need not be the same. Only the product of the chosen two factors and their reverses must equal k and R(k) in the given bases. See the example below and the linked data file.

No numbers are currently known that have solutions in seven or more bases. Assuming a(13) exists it is greater than 10^9.

Examples

			1122 is a term as k = A * B and R(k) = R(A) * R(B) has solutions in the six bases 4,5,7,8,9,10. See the table below for k = 1122.
.
      base   | k_base | A_base * B_base | R(k_base) | R(A_base) * R(B_base)
  =========================================================================
       4     | 101202 |    101 * 1002   |  202101   |       101 * 2001
  in base 10 |   1122 |     17 * 66     |    2193   |        17 * 129
  ------------------------------------------------------------------------
       5     |  13442 |      3 * 2444   |   24431   |         3 * 4442
  in base 10 |   1122 |      3 * 374    |    1866   |         3 * 622
  ------------------------------------------------------------------------
       7     |   3162 |     31 * 102    |    2613   |        13 * 201
  in base 10 |   1122 |     22 * 51     |     990   |        10 * 99
  -------------------------------------------------------------------------
       8     |   2142 |     21 * 102    |    2412   |        12 * 201
  in base 10 |   1122 |     17 * 66     |    1290   |        10 * 129
  -------------------------------------------------------------------------
       9     |   1476 |     12 * 123    |    6741   |        21 * 321
  in base 10 |   1122 |     11 * 102    |    4978   |        19 * 262
  -------------------------------------------------------------------------
      10     |   1122 |     11 * 102    |    2211   |        11 * 201
.
The bases used in the twelve terms below 10^9 are as follows:
.
         k    |       bases
  --------------------------------
        1122  |  4, 5, 7, 8, 9, 10
       17875  |  2, 3, 4, 6, 8, 10
       65331  |  2, 4, 5, 6, 8, 10
      367598  |  3, 4, 6, 8, 9, 10
      818545  |  2, 3, 4, 6, 8,  9
     1997905  |  2, 3, 4, 6, 8,  9
    43998955  |  2, 3, 4, 8, 9, 10
   100383283  |  2, 3, 4, 6, 9, 10
   112887775  |  2, 3, 4, 8, 9, 10
   112977865  |  2, 3, 4, 8, 9, 10
   145683265  |  2, 3, 4, 6, 8,  9
   230034805  |  2, 3, 4, 6, 8,  9
.
		

Links

• Scott R. Shannon, Factorizations of the twelve terms below 10^9.

Crossrefs

Cf. A346133, A066531, A230594, A001055, A038548.

Extensions

a(13) from Michael S. Branicky, Jun 21 2023