OFFSET
0,3
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.
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
EXAMPLE
a(29) = 2021 as 29 = 2*12 + 0*6 + 2*2 + 1*1.
PROG
(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
{\\ 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
CROSSREFS
KEYWORD
base,easy,nonn
AUTHOR
Rick L. Shepherd, Jul 03 2005
STATUS
approved