cp's OEIS Frontend

Exercise 15 on page 30 of the Long textbook is "Let m_1, m_2, m_3, ... and M_0, M_1, M_2, ... be as above. [see A109827.] Let s_0, s_1, s_2, ... be an infinite sequence of zeros and ones containing infinitely many of each. Show that *every* integer r (positive, negative, or zero) can be represented uniquely in the form r = (-1)^s_n c_n M_n + (-1)^s_(n-1) c_(n-1) M_(n-1) + ... + (-1)^s_1 c_1 M_1 + (-1)^s_0 c_0 M_0 where c_n <> 0 for r <> 0 and 0 <= c_i < m_(i+1) for all i. If r is positive show that s_n = 0 and if r is negative show that s_n = 1." Take the primes (A000040) for the m_i. Then the M_i are the primorials (A002110). Take the binary expansion of Pi (A004601) for the s_k. This sequence, a(r) = (c_n c_(n-1) ... c_1 c_0 concatenated), gives the representations of the nonnegative integers. See A109839 for the corresponding negative integers.