Rebasing notation

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 is 2[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 to b[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 like GF2[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)
• $n_{b->q}$ (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:
  • b[n]1 gives the sum of base-b digits; for example 2[n]1 is A000120, 10[n]1 is A007953
  • b[n]0 arguably* gives n mod b, the least-significant base-b digit (* depending on your 0^0 religion)
  • b[n](-1) gives the alternating base-b digit sum; for example 2[n](-1) is A065359, etc.
• 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:

• A065359
• A065363, A065364, A065365, A065366, A065367, A065368, A065369 (balanced ternary)
• A065720, A065721, A065722, A065723, A065724, A065725, A065726, A065727 (rebasing maintains prime property)
• A066321, A122029, A122621, A164702, A175245, A190820, A215160, A235265, A235354