0,3

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.

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.

Calvin T. Long, Elementary Introduction to Number Theory, 2nd ed., D.C. Heath and Company, 1972, p. 30.

a(29) = 2021 as 29 = 2*12 + 0*6 + 2*2 + 1*1.

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).

Sequence in context: A165265 A049345 A007623 this_sequence A109839 A087486 A102626

Adjacent sequences: A109824 A109825 A109826 this_sequence A109828 A109829 A109830

base,easy,nonn

Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 03 2005