A109827 Numbers written in an alternating binary-then-ternary base.

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

Offset: 0

Author

Rick L. Shepherd, Jul 03 2005

Keywords

• base
• easy
• nonn

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 = 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.

Examples

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

References

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

Links

• Rick L. Shepherd, Table of n, a(n) for n = 0..9999

Crossrefs

Cf. A010693 (2, 3, 2, 3, ...), A026549 (place values), A089293 (sum of digits).

Cf. A055643 (Babylonian numbers), A007623 (numbers in factorial base), A049345 (numbers in primorial base), A007088 (numbers in base 2: binary), A007089 (numbers in base 3: ternary).

Programs

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)
}
a(n) = b(n, A010693) \\ Rick L. Shepherd, Aug 04 2021

Python

a109827 = lambda n: 100 * a109827(n // 6) + 10 * ((n % 6) // 2) + n % 2 if n else 0 # David Radcliffe, Aug 03 2021