0,2

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.

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

a(13) = 321 as 13 = 18 - 4 - 1 = ((-1)^0)*3*6 + ((-1)^1)*2*2 + ((-1)^1)*1*1, where 1,1,0 are the first three terms of A004601 and 1,2,6 are the first three terms of A002110.

Cf. A109839 (negative numbers represented similarly), A109827, A004601 (Pi in binary), A000040 (primes), A002110 (primorials), A007623 (factorial base).

Sequence in context: A014756 A014748 A152144 this_sequence A014734 A137517 A014735

Adjacent sequences: A109835 A109836 A109837 this_sequence A109839 A109840 A109841

base,nonn

Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 04 2005