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Rebasing notation
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Contents
b[n]q
: rebase from base b into base q
In 2005, Marc LeBrun described the rebasing notation (cf. A000695):
- This may be described concisely using the "rebase" notation
b[n]q
, which means "replace b with q in the expansion of n", thus "rebasing" n from base b into base q. The present sequence is2[n]4
. Many interesting operations (e.g.,10[n](1/10)
= digit reverse, A004086, shifted by a suitable power of 10) are nicely expressible this way.
- Note that
q[n]b
is (roughly) inverse tob[n]q
.
- It's also natural to generalize the idea of "basis" so as to cover the likes of
F[n]2
, the so-called "fibbinary" numbers (A003714) and provide standard ready-made images of entities obeying other arithmetics, say likeGF2[n]2
(e.g., primes = A014580, squares = the present sequence, etc.).
The following table shows some rebasing-related sequences in the OEIS:
b=2 | b=3 | b=4 | b=5 | b=6 | b=7 | b=8 | b=9 | b=10 | |
---|---|---|---|---|---|---|---|---|---|
q=2 | A065361 | A0653622 | A215088 | A215089 | A203580 | A028897 | |||
q=3 | A0058361 | A215090 | A2150922 | A028898 | |||||
q=4 | A000695 | A023717 | A303787 | A028899 | |||||
q=5 | A033042 | A0374532 | A0374592 | A303788 | A028900 | ||||
q=6 | A033043 | A037454 | A037460 | A037465 | A303789 | A028901 | |||
q=7 | A0330441 | A037455 | A0374612 | A037466 | A037470 | A028902 | |||
q=8 | A033045 | A037456 | A037462 | A037467 | A037471 | A037474 | A028903 | ||
q=9 | A033046 | A037463 | A037468 | A037472 | A037475 | A037477 | A028904 | ||
q=10 | A007088 | A007089 | A007090 | A007091 | A007092 | A007093 | A007094 | A007095 A0374792 |
|
q=11 | A033047 | ||||||||
q=12 | A033048 | A1024871 | |||||||
q=13 | A033049 | A094823 | |||||||
q=14 | A033050 | ||||||||
q=15 | A033051 | ||||||||
q=16 | A033052 | A102489 | |||||||
q=17 | A197351 | ||||||||
q=18 | A197352 | ||||||||
q=19 | A197353 | ||||||||
q=20 | A063012 | A1024911 |
1 These sequences have offset 1 and start with n=0.
2 These sequences have offset 1 and start with n=1.
All other sequences have offset 0 and start with n=0.
Notational variants
Sometimes variants of the notation are used in the OEIS:
- A065361:
(3)[n](2)
- (LaTeX).
Sums of distinct powers of q
The first column (b=2) of the table above shows the sequences for Sums of distinct powers of q, since the binary digits in n enumerate all such powers.
Examples
A037454: 3[n]6 n = 0 1 2 3 4 5 6 7 8 9 10 11 a(n) = 0, 1, 2, 6, 7, 8, 12, 13, 14, 36, 37, 38, ... n = 11: 1110 = 1023 -> 1026 = 1*6^2 + 0*6^1 + 2*6^0 = 3810 = a(11)
Programs
- Maple (after R. J. Mathar in A215160):
rebase := proc(bfrom, n, bto) local c, i; c := convert(n, base, bfrom) ; add(op(i, c)*bto^(i-1), i=1..nops(c)) ; end proc: seq(rebase(3,n,6),n=0..11);
- (Mathematica)
b:=3; q:=6; Table[FromDigits[RealDigits[n, b], q], {n, 0, 100}]
- (PARI)
b=3; q=6; for(n=0,100,print1(fromdigits(digits(n, b), q),","));
- Recursive definition (pseudo-Python, Marc LeBrun):
if n == 0 return 0 else return q * rebase(b, floor(n / b), q) + (n % b)
- Java (jOEIS)
java -cp joeis.jar irvine.oeis.a037.A037454 3 6
Other applications
Marc LeBrun points out:
- There are some interesting columns with q < 2 which could be added to the table above:
- There may even be sequences with b<0 as well.
- f(x) = b[x]q for real x with fixed b and q is an interesting function.
Also related are: