A109838 - OEIS

%I #2 Mar 30 2012 17:36:43

%S 0,121,120,111,110,101,100,221,220,211,210,201,200,321,320,311,310,

%T 301,300,421,420,411,410,401,400,1021,1020,1011,1010,1001,1000,1121,

%U 1120,1111,1110,1101,1100,1221,1220,1211,1210,1201,1200,1321,1320,1311,1310

%N Numbers written in a bits-of-Pi/primorial base system.

%C 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.

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

%e 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.

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

%K base,nonn

%O 0,2

%A _Rick L. Shepherd_, Jul 04 2005