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A109827
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Numbers written in an alternating binary-then-ternary base.
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5
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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
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OFFSET
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0,3
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COMMENTS
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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 = Product_{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
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Calvin T. Long, Elementary Introduction to Number Theory, 2nd ed., D.C. Heath and Company, 1972, p. 30.
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LINKS
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EXAMPLE
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a(29) = 2021 as 29 = 2*12 + 0*6 + 2*2 + 1*1.
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PROG
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(Python) a109827 = lambda n: 100 * a109827(n // 6) + 10 * ((n % 6) // 2) + n % 2 if n else 0 # David Radcliffe, Aug 03 2021
(PARI) my(table=[0, 1, 10, 11, 20, 21]); a(n) = fromdigits(apply(d->table[d+1], digits(n, 6)), 100); \\ Kevin Ryde, Aug 03 2021
(PARI) A010693(n) = if(n%2, 2, 3) \\ Function m is A010693 with index 1 here.
{\\ The function b(n, m) works for all nonnegative n and every sequence m of (mixed or constant) radices as described above.
my(c, d, k, ntmp, p, v, x); b(n, m) = if(n < 0, , v = [1]; k = 0;
while(1, k++; p = v[#v]*m(k); if(p <= n, v = concat(v, p), break));
ntmp = n; c = [];
forstep(i = #v, 1, -1, d = ntmp\v[i]; c = concat(c, d); ntmp = ntmp - d*v[i]);
x = 10; if(vecmax(c) < x, eval(Pol(c, 'x)), c))
\\ returned value is a vector of decimal coefficients if any calculated
\\ digit is larger than 9 (i.e., not suitable as an OEIS term)
}
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CROSSREFS
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Cf. A010693 (2, 3, 2, 3, ...), A026549 (1, 2, 6, 12, 36, ...), A007623 (numbers in factorial base), A049345 (numbers in primorial base), A007088 (numbers in base 2: binary), A007089 (numbers in base 3: ternary), A089293 (sum of digits).
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KEYWORD
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base,easy,nonn
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AUTHOR
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STATUS
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approved
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