|
| |
|
|
A109838
|
|
Numbers written in a bits-of-Pi/primorial base system.
|
|
1
| |
|
|
0, 121, 120, 111, 110, 101, 100, 221, 220, 211, 210, 201, 200, 321, 320, 311, 310, 301, 300, 421, 420, 411, 410, 401, 400, 1021, 1020, 1011, 1010, 1001, 1000, 1121, 1120, 1111, 1110, 1101, 1100, 1221, 1220, 1211, 1210, 1201, 1200, 1321, 1320, 1311, 1310
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| 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.
|
|
|
REFERENCES
| Calvin T. Long, Elementary Introduction to Number Theory, 2nd ed., D.C. Heath and Company, 1972, p. 30.
|
|
|
EXAMPLE
| 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.
|
|
|
CROSSREFS
| Cf. A109839 (negative numbers represented similarly), A109827, A004601 (Pi in binary), A000040 (primes), A002110 (primorials), A007623 (factorial base).
Sequence in context: A197817 A152144 A202003 * A014734 A137517 A014735
Adjacent sequences: A109835 A109836 A109837 * A109839 A109840 A109841
|
|
|
KEYWORD
| base,nonn
|
|
|
AUTHOR
| Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 04 2005
|
| |
|
|
