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A109827
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Numbers written in an alternating binary-then-ternary base.
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1
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0, 1, 10, 11, 20, 21, 100, 101, 110, 111, 120, 121, 1000, 1001, 1010, 1011, 1020, 1021, 1100, 1101, 1110, 1111, 1120, 1121, 2000, 2001, 2010, 2011, 2020, 2021, 2100, 2101, 2110, 2111, 2120, 2121, 10000, 10001, 10010, 10011, 10020, 10021, 10100, 10101
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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.
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REFERENCES
| Calvin T. Long, Elementary Introduction to Number Theory, 2nd ed., D.C. Heath and Company, 1972, p. 30.
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EXAMPLE
| a(29) = 2021 as 29 = 2*12 + 0*6 + 2*2 + 1*1.
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CROSSREFS
| 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: A197652 A049345 A007623 * A109839 A087486 A102626
Adjacent sequences: A109824 A109825 A109826 * A109828 A109829 A109830
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KEYWORD
| base,easy,nonn
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AUTHOR
| Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 03 2005
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