%I #20 Apr 25 2024 13:32:19
%S 297,440,1440,8596,15552,121638,282240,1039104,5757696,24108000
%N The smallest number k which, when written in base n, has a factorization k = f1*f2*...*fr where the digits of {k, f1, f2, ..., fr} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.
%C The offset begins at 7 as in bases 2..6 no term exists.
%C Assumes all factors f_i > 1. If f_i = 1 is allowed, then a(7) = 104 = 1*4*26, a(10) = 4830 = 1*2*5*7*69, a(12) = 72240 = 1*4*6*7*430, a(15) = 4244940 = 1*2*3*7*9*10*1123, ... - _Chai Wah Wu_, Apr 24 2024
%e The terms and their factorizations are:
%e a(7) = 297 = [9, 33] = 603_7 = [12_7, 45_7] = "6031245" which contains all digits 0..6 once.
%e a(8) = 440 = [2, 4, 5, 11] = 670_8 = [2_8, 4_8, 5_8, 13_8] = "67024513" which contains all digits 0..7 once.
%e a(9) = 1440 = [3, 4, 5, 24] = 1870_9 = [3_9, 4_9, 5_9, 26_9] = "187034526" which contains all digits 0..8 once.
%e a(10) = 8596 = [2, 14, 307] = "8596214307" which contains all digits 0..9 once. See also A370970.
%e a(11) = 15552 = [2, 3, 6, 8, 54] = 10759_11 = [2_11, 3_11, 6_11, 8_11, 4a_11] = "1075923684a" which contains all digits 0..a once.
%e a(12) = 121638 = [2, 3, 11, 1843] = 5a486_12 = [2_12, 3_12, b_12, 1097_12] = "5a48623b1097" which contains all digits 0..b once.
%e a(13) = 282240 = [2, 3, 5, 7, 21, 64] = 9b60a_13 = [2_13, 3_13, 5_13, 7_13, 18_13, 4c_13] = "9b60a2357184c" which contains all digits 0..c once.
%e a(14) = 1039104 = [2, 3, 4, 6, 8, 11, 82] = 1d097a_14 = [2_14, 3_14, 4_14, 6_14, 8_14, b_14, 5c_14] = "1d097a23468b5c" which contains all digits 0..d once.
%e From _Chai Wah Wu_, Apr 24 2024: (Start)
%e a(15) = 5757696 = [2, 3, 4, 12, 84, 238] = 78aeb6_15 = [2_15, 3_15, 4_15, c_15, 59_15, 10d_15] = "78aeb6234c5910d" which contains all digits 0..e once.
%e a(16) = 24108000 = [3, 4, 5, 7, 10, 41, 140] = 16fdbe0_16 = [3_16, 4_16, 5_16, 7_16, a_16, 29_16, 8c_16] = "16fdbe03457a298c" which contains all digits 0..f once. (End)
%Y Cf. A372053, A370970, A371958.
%K nonn,base,more
%O 7,1
%A _Scott R. Shannon_, Apr 24 2024
%E a(15)-a(16) from _Chai Wah Wu_, Apr 24 2024
%E Added escape clause to definition at the suggestion of _Chai Wah Wu_. - _N. J. A. Sloane_, Apr 25 2024