%I A109827
%S A109827 0,1,10,11,20,21,100,101,110,111,120,121,1000,1001,1010,1011,1020,1021,
%T A109827 1100,1101,1110,1111,1120,1121,2000,2001,2010,2011,2020,2021,2100,2101,
%U A109827 2110,2111,2120,2121,10000,10001,10010,10011,10020,10021,10100,10101
%N A109827 Numbers written in an alternating binary-then-ternary base.
%C A109827 Exercise 14 on page 30 of the Long textbook is "Let m_1, m_2, m_3 ... 
               be an infinite sequence of integers such that m_i >= 2 for all i. 
               Let M_0 = 1 and M_i = prod(j=1,...,i) m_j for all i >= 1. Show that 
               every nonnegative integer r can be written uniquely in the form r 
               = c_n M_n + c_(n-1) M_(n-1) + ... + c_1 M_1 + c_0 where c_n <> 0 
               for r <> 0 and 0 <= c_i < m_(i+1) for all i." The current sequence 
               of terms a(r) = (c_n c_(n-1) ... c_1 c_0 concatenated) is one example 
               of an infinite family of hybrid representations (just using only 
               2 and 3). For the m_i, this sequence uses A010693. Then the corresponding 
               M_i are A026549. Thus the places reading from right have values (1,
               2,2*3,2*3*2,2*3*2*3,...) = A026549. The (ternary) digit 2 may only 
               appear in the even positions counting from the rightmost as position 
               1. Appending "00" to any term multiplies the number by 6.
%C A109827 However, appending a single "0" to a term multiplies the number by 2 
               or by 3 or produces an invalid string of digits -- or even none of 
               the above (110 => 1100, 8 becomes 18) -- depending upon the original 
               number and its length.
%D A109827 Calvin T. Long, Elementary Introduction to Number Theory, 2nd ed., D.C. 
               Heath and Company, 1972, p. 30.
%e A109827 a(29) = 2021 as 29 = 2*12 + 0*6 + 2*2 + 1*1.
%Y A109827 Cf. A010693 (2, 3, 2, 3, ...), A026549 (1, 2, 6, 12, 36, ...), A049345 
               (numbers in primorial base), A007088 (numbers in base 2: binary), 
               A007089 (numbers in base 3: ternary).
%Y A109827 Sequence in context: A165265 A049345 A007623 this_sequence A109839 A087486 
               A102626
%Y A109827 Adjacent sequences: A109824 A109825 A109826 this_sequence A109828 A109829 
               A109830
%K A109827 base,easy,nonn
%O A109827 0,3
%A A109827 Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 03 2005