%I #39 Jun 22 2023 01:10:55
%S 1122,17875,65331,367598,818545,1997905,43998955,100383283,112887775,
%T 112977865,145683265,230034805,5231187650
%N 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.
%C 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.
%C 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.
%C No numbers are currently known that have solutions in seven or more bases. Assuming a(13) exists it is greater than 10^9.
%H Scott R. Shannon, <a href="/A346219/a346219_1.txt">Factorizations of the twelve terms below 10^9</a>.
%e 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.
%e .
%e base | k_base | A_base * B_base | R(k_base) | R(A_base) * R(B_base)
%e =========================================================================
%e 4 | 101202 | 101 * 1002 | 202101 | 101 * 2001
%e in base 10 | 1122 | 17 * 66 | 2193 | 17 * 129
%e ------------------------------------------------------------------------
%e 5 | 13442 | 3 * 2444 | 24431 | 3 * 4442
%e in base 10 | 1122 | 3 * 374 | 1866 | 3 * 622
%e ------------------------------------------------------------------------
%e 7 | 3162 | 31 * 102 | 2613 | 13 * 201
%e in base 10 | 1122 | 22 * 51 | 990 | 10 * 99
%e -------------------------------------------------------------------------
%e 8 | 2142 | 21 * 102 | 2412 | 12 * 201
%e in base 10 | 1122 | 17 * 66 | 1290 | 10 * 129
%e -------------------------------------------------------------------------
%e 9 | 1476 | 12 * 123 | 6741 | 21 * 321
%e in base 10 | 1122 | 11 * 102 | 4978 | 19 * 262
%e -------------------------------------------------------------------------
%e 10 | 1122 | 11 * 102 | 2211 | 11 * 201
%e .
%e The bases used in the twelve terms below 10^9 are as follows:
%e .
%e k | bases
%e --------------------------------
%e 1122 | 4, 5, 7, 8, 9, 10
%e 17875 | 2, 3, 4, 6, 8, 10
%e 65331 | 2, 4, 5, 6, 8, 10
%e 367598 | 3, 4, 6, 8, 9, 10
%e 818545 | 2, 3, 4, 6, 8, 9
%e 1997905 | 2, 3, 4, 6, 8, 9
%e 43998955 | 2, 3, 4, 8, 9, 10
%e 100383283 | 2, 3, 4, 6, 9, 10
%e 112887775 | 2, 3, 4, 8, 9, 10
%e 112977865 | 2, 3, 4, 8, 9, 10
%e 145683265 | 2, 3, 4, 6, 8, 9
%e 230034805 | 2, 3, 4, 6, 8, 9
%e .
%Y Cf. A346133, A066531, A230594, A001055, A038548.
%K nonn,base,more
%O 1,1
%A _Scott R. Shannon_, _Eric Angelini_ and _Carole Dubois_, Jul 11 2021
%E a(13) from _Michael S. Branicky_, Jun 21 2023