A024629 -id:A024629 - cp's OEIS frontend

A081848 Number of numbers whose base-3/2 expansion (see A024629) has n digits.

3, 3, 3, 6, 9, 12, 18, 27, 42, 63, 93, 141, 210, 315, 474, 711, 1065, 1599, 2397, 3597, 5394, 8091, 12138, 18207, 27309, 40965, 61446, 92169, 138255, 207381, 311073, 466608, 699912, 1049868, 1574802, 2362203, 3543306, 5314959, 7972437, 11958657

Offset: 1

N. J. A. Sloane, Apr 13 2003

Run lengths in A246435. - Reinhard Zumkeller, Sep 05 2014

			a(1) = 3 because 0, 1 and 2 each have 1 digit.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its sequences, arXiv:1808.04304 [math.NT], 2018.
J. S. Tanton, A collection of research problems.

Cf. A024629, A070885, A073941, A246435.

a081848 n = a081848_list !! (n-1)
a081848_list = 3 : tail (zipWith (-) (tail a070885_list) a070885_list)
-- Reinhard Zumkeller, Sep 05 2014

from itertools import islice
def A081848_gen(): # generator of terms
    yield (a:=3)
    while True:
        yield (b:=(a+1>>1)+(a&1))
        a += b
A081848_list = list(islice(A081848_gen(),70)) # Chai Wah Wu, Sep 20 2022

For n > 1, a(n) = A070885(n+1) - A070885(n). - Tom Edgar, Jun 25 2014

a(n) = 3*A073941(n). - Tom Edgar, Jul 21 2014

More terms from David Wasserman, Jun 28 2004

Edited by Charles R Greathouse IV, Aug 02 2010

A304273 The concatenation of the first n terms is the smallest positive even number with n digits when written in base 3/2 (cf. A024629).

Original entry on oeis.org

2, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 09 2018

This sequence exists since the smallest even integers (see A303500) are prefixes of each other.

Apparently a variant of A205083. - R. J. Mathar, Jun 09 2018

			The number 5 in base 3/2 is 22, and the number 6 is 210. Therefore 210 is the smallest even integer with 3 digits in base 3/2. Its prefix 21 is 4: the smallest even integer with 2 digits in base 3/2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.

Cf. A005428, A070885, A073941, A081848, A024629, A246435, A304024, A304025, A303500, A304272, A304274.

A005428 a(n) = ceiling((1 + sum of preceding terms) / 2) starting with a(0) = 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 14, 21, 31, 47, 70, 105, 158, 237, 355, 533, 799, 1199, 1798, 2697, 4046, 6069, 9103, 13655, 20482, 30723, 46085, 69127, 103691, 155536, 233304, 349956, 524934, 787401, 1181102, 1771653, 2657479, 3986219, 5979328, 8968992, 13453488, 20180232, 30270348, 45405522, 68108283, 102162425, 153243637, 229865456, 344798184

Offset: 0

N. J. A. Sloane and Simon Plouffe

Original definition: a(0) = 1, state(0) = 2; for n >= 1, if a(n-1) is even then a(n) = 3*a(n-1)/2 and state(n) = state(n-1); if a(n-1) is odd and state(n-1) = 1 then a(n) = ceiling( 3*a(n-1)/2) and state(n) = 3 - state(n-1) and if a(n-1) is odd and state(n-1) = 2 then a(n) = floor( 3*a(n-1)/2) and state(n) = 3 - state(n-1). [See formula by M. Alekseyev for a simpler equivalent. - Ed.]
Arises from a version of the Josephus problem: sequence gives set of n where, if you start with n people and every 3rd person drops out, either it is person #1 or #2 who is left at the end. A081614 and A081615 give the subsequences where it is person #1 (respectively #2) who is left.
The state changes just when a(n) is odd: it therefore indicates whether the sum of a(0) to a(n) is odd (1 means no, 2 means yes).
The sum a(0) to a(n) is never divisible by 3 (for n >= 0); it is 1 mod 3 precisely when the sum a(0) to a(n-1) is odd and thus indicates the state at the previous step. - Franklin T. Adams-Watters, May 14 2008
The number of nodes at level n of a planted binary tree with alternating branching and non-branching nodes. - Joseph P. Shoulak, Aug 26 2012
Take Sum_{k=1..n} a(k) objects, and partition them into 3 parts. It is always possible to generate those parts using addends once each from the initial n terms, and this is the fastest growing sequence with this property. For example, taking 1+1+2+3+4+6+9 = 26 objects, if we partition them [10,9,7], we can generate these sizes as 10 = 9+1, 9 = 6+3, 7 = 4+2+1. The corresponding sequence partitioning into 2 parts is the powers of 2, A000079. In general, to handle partitioning into k parts, replace the division by 2 in the definition with division by k-1. - Franklin T. Adams-Watters, Nov 07 2015
a(n) is the number of even integers that have n+1 digits when written in base 3/2. For example, there are 2 even integers that use three digits in base 3/2: 6 and 8: they are written as 210 and 212, respectively. - Tanya Khovanova and PRIMES STEP Senior group, Jun 03 2018

			n........0...1...2...3...4...5...6...7...8...9..10..11..12..13..14.
state=1......1...........4...6...9..........31.....70..105.........
state=2..1.......2...3..............14..21......47.........158..237

F. Schuh, The Master Book of Mathematical Recreations. Dover, NY, 1968, page, 374, Table 18, union of columns 1 and 2 (which are A081614 and A081615).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
K. Burde, Das Problem der Abzählreime und Zahlentwicklungen mit gebrochenen Basen [The problem of counting rhymes and number expansions with fractional bases], J. Number Theory 26(2) (1987), 192-209.
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its sequences, arXiv:1808.04304 [math.NT], 2018.
L. Halbeisen and N. Hungerbuehler, The Josephus Problem, J. Théor. Nombres Bordeaux 9(2) (1997), 303-318.
A. M. Odlyzko and H. S. Wilf, Functional iteration and the Josephus problem, Glasgow Math. J. 33 (1991), 235-240.
Eric Weisstein's World of Mathematics, Josephus Problem.
Wikipedia, Josephus problem.
Index entries for sequences related to the Josephus Problem

Cf. A005427, A073941, A082416. Union of A081614 and A081615.
First differences of D_3(n) (A061419) in the terminology of Odlyzko and Wilf. - Ralf Stephan, Apr 23 2002
Same as log_2(A082125(n+3)). - Ralf Stephan, Apr 16 2002
Apart from initial terms, same as A073941, which has further information.
a(n) is the number of positive even k for which A024629(k) has n+1 digits. - Glen Whitney, Jul 09 2017
Partial sums are in A061419, A061418, A006999.
Cf. A000079, A072493, A120160, A120170, A120178, A120186, A120194, A120202.

a005428 n = a005428_list !! n
a005428_list = (iterate j (1, 1)) where
   j (a, s) = (a', (s + a') `mod` 2) where
     a' = (3 * a + (1 - s) * a `mod` 2) `div` 2
-- Reinhard Zumkeller, May 10 2015 (fixed), Oct 26 2011

f[s_] := Append[s, Ceiling[(1 + Plus @@ s)/2]]; Nest[f, {1}, 40] (* Robert G. Wilson v, Jul 07 2006 *)
nxt[{t_,a_}]:=Module[{c=Ceiling[(1+t)/2]},{t+c,c}]; NestList[nxt,{1,1},50][[All,2]] (* Harvey P. Dale, Nov 05 2017 *)

{ a=1; s=2; for(k=1,50, print1(a,", "); a=(3*a+s-1)\2; s=(s+a)%3; ) } \\ Max Alekseyev, Mar 28 2009

s=0;vector(50,n,-s+s+=s\2+1)  \\ M. F. Hasler, Oct 14 2012

from itertools import islice
def A005428_gen(): # generator of terms
    a, c = 1, 0
    yield 1
    while True:
        yield (a:=1+((c:=c+a)>>1))
A005428_list = list(islice(A005428_gen(),30)) # Chai Wah Wu, Sep 21 2022

a(0) = 1; a(n) = ceiling((1 + Sum_{k=0..n-1} a(k)) / 2). - Don Reble, Apr 23 2003
a(1) = 1, s(1) = 2, and for n > 1, a(n) = floor((3*a(n-1) + s(n-1) - 1) / 2), s(n) = (s(n-1) + a(n)) mod 3. - Max Alekseyev, Mar 28 2009
a(n) = floor(1 + (sum of preceding terms)/2). - M. F. Hasler, Oct 14 2012

More terms from Hans Havermann, Apr 23 2003

Definition replaced with a simpler formula due to Don Reble, by M. F. Hasler, Oct 14 2012

A244040 Sum of digits of n in fractional base 3/2.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 4, 5, 3, 4, 5, 5, 6, 7, 4, 5, 6, 5, 6, 7, 7, 8, 9, 5, 6, 7, 5, 6, 7, 7, 8, 9, 8, 9, 10, 5, 6, 7, 7, 8, 9, 6, 7, 8, 7, 8, 9, 9, 10, 11, 9, 10, 11, 5, 6, 7, 7, 8, 9, 8, 9, 10, 6, 7, 8, 8, 9, 10, 8, 9, 10, 9, 10, 11, 11, 12, 13, 10, 11, 12, 5

Offset: 0

James Van Alstine, Jun 17 2014

The base 3/2 expansion is unique, and thus the sum of digits function is well-defined.

Fixed point starting with 0 of the two-block substitution a,b -> a,a+1,a+2 for a = 0,1,2,... and b = 0,1,2,.... - Michel Dekking, Sep 29 2022

			In base 3/2 the number 7 is represented by 211 and so a(7) = 2 + 1 + 1 = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Michel Dekking, The Thue-Morse sequence in base 3/2, J. Int. Seq., Vol. 26 (2023), Article 23.2.3; arXiv preprint, arXiv:2301.13563 [math.CO], 2023.
Index entries for sequences related to fractional bases.

Cf. A024629, A007953, A000120, A053735, A244041, A246435.

a244040 0 = 0
a244040 n = a244040 (2 * n') + t where (n', t) = divMod n 3
-- Reinhard Zumkeller, Sep 05 2014

a[n_]:= a[n]= If[n==0, 0, a[2*Floor[n/3]] + Mod[n,3]]; Table[a[n], {n, 0, 85}] (* G. C. Greubel, Aug 20 2019 *)

a(n) = if(n == 0, 0, a(n\3 * 2) + n % 3); \\ Amiram Eldar, Jul 30 2025

a244040 = lambda n: a244040((n // 3) * 2) + (n % 3) if n else 0 # David Radcliffe, Aug 21 2021

def base32sum(n):
    L, i = [n], 1
    while L[i-1]>2:
        x=L[i-1]
        L[i-1]=x.mod(3)
        L.append(2*floor(x/3))
        i+=1
    return sum(L)
[base32sum(n) for n in [0..85]]

a(0) = 0, a(3n+r) = a(2n)+r for n >= 0 and r = 0, 1, 2. - David Radcliffe, Aug 21 2021

a(n) = A007953(A024629(n)). - Amiram Eldar, Jul 30 2025

A342426 Niven numbers in base 3/2: numbers divisible by their sum of digits in fractional base 3/2 (A244040).

Original entry on oeis.org

1, 2, 6, 9, 14, 21, 40, 42, 56, 72, 84, 108, 110, 120, 126, 130, 143, 154, 156, 162, 165, 168, 169, 176, 180, 182, 189, 198, 220, 225, 231, 243, 252, 280, 288, 297, 306, 308, 320, 322, 330, 336, 348, 350, 364, 390, 423, 430, 432, 459, 460, 462, 480, 490, 504

Offset: 1

Amiram Eldar, Mar 11 2021

			6 is a term since its representation in base 3/2 is 210 and 2 + 1 + 0 = 3 is a divisor of 6.
9 is a term since its representation in base 3/2 is 2100 and 2 + 1 + 0 + 0 = 3 is a divisor of 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A024629, A244040.
Subsequences: A342427, A342428, A342429.
Similar sequences: A005349 (decimal), A049445 (binary), A064150 (ternary), A064438 (quaternary), A064481 (base 5), A118363 (factorial), A328208 (Zeckendorf), A328212 (lazy Fibonacci), A331085 (negaFibonacci), A333426 (primorial), A334308 (base phi), A331728 (negabinary).

s[0] = 0; s[n_] := s[n] = s[2*Floor[n/3]] + Mod[n, 3]; q[n_] := Divisible[n, s[n]]; Select[Range[500], q]

A246435 Length of representation of n in fractional base 3/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9

Offset: 0

Reinhard Zumkeller, Sep 05 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its sequences, arXiv:1808.04304 [math.NT], 2018.
Index entries for sequences related to fractional bases.

Cf. A024629, A055642, A070989, A081604, A081848 (run lengths), A244040.

a246435 n = if n < 3 then 1 else a246435 (2 * div n 3) + 1
-- Reinhard Zumkeller, Sep 05 2014

a[n_] := If[n < 3, 1, a[2 Quotient[n, 3]] + 1]; Array[a, 100, 0] (* Jean-François Alcover, Feb 05 2019 *)

a(n) = if(n < 3, 1, a(n\3 * 2) + 1); \\ Amiram Eldar, Jul 30 2025

a(n) = if n < 3 then 1, otherwise a(2*floor(n/3)) + 1.

a(n) = A055642(A024629(n)).

A070885 a(n) = (3/2)a(n-1) if a(n-1) is even; (3/2)(a(n-1)+1) if a(n-1) is odd.

Original entry on oeis.org

1, 3, 6, 9, 15, 24, 36, 54, 81, 123, 186, 279, 420, 630, 945, 1419, 2130, 3195, 4794, 7191, 10788, 16182, 24273, 36411, 54618, 81927, 122892, 184338, 276507, 414762, 622143, 933216, 1399824, 2099736, 3149604, 4724406, 7086609, 10629915

Offset: 1

Eric W. Weisstein, May 14 2002

nonn

The smallest positive number such that A024629(a(n)) has n digits, per page 9 of the Tanton reference in Links. - Glen Whitney, Sep 17 2017

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, 2002, p. 123.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.
James Tanton, Exploding Dots, Chapter 9
Eric Weisstein's World of Mathematics, Wolfram Sequences

The constant K is 2/3*K(3) (see A083286). - Ralf Stephan, May 29 2003
Cf. A003312.
Cf. A081848.
Cf. A205083 (parity of terms).

a070885 n = a070885_list !! (n-1)
a070885_list = 1 : map (flip (*) 3 . flip div 2 . (+ 1)) a070885_list
-- Reinhard Zumkeller, Sep 05 2014

A070885 := proc(n)
    option remember;
    if n = 1 then
        return 1;
    elif type(procname(n-1),'even') then
        procname(n-1) ;
    else
        procname(n-1)+1 ;
    end if;
    %*3/2 ;
end proc:
seq(A070885(n),n=1..80) ; # R. J. Mathar, Jun 18 2018

NestList[If[EvenQ[#],3/2 #,3/2 (#+1)]&,1,40] (* Harvey P. Dale, May 18 2018 *)

from itertools import islice
def A070885_gen(): # generator of terms
    a = 1
    while True:
        yield a
        a += (a+1>>1)+(a&1)
A070885_list = list(islice(A070885_gen(),70)) # Chai Wah Wu, Sep 20 2022

For n > 1, a(n) = 3*A061419(n) = 3*floor(K*(3/2)^n) where K=1.08151366859... - Benoit Cloitre, Aug 18 2002
a(n) = 3*ceiling(a(n-1)/2). - Benoit Cloitre, Apr 25 2003
a(n+1) = a(n) + A081848(n), for n > 1. - Reinhard Zumkeller, Sep 05 2014

A087088 Positive ruler-type fractal sequence with 1's in every third position.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 1, 5, 3, 1, 2, 6, 1, 4, 2, 1, 3, 7, 1, 2, 5, 1, 3, 2, 1, 4, 8, 1, 2, 3, 1, 6, 2, 1, 4, 3, 1, 2, 5, 1, 9, 2, 1, 3, 4, 1, 2, 7, 1, 3, 2, 1, 5, 4, 1, 2, 3, 1, 6, 2, 1, 10, 3, 1, 2, 4, 1, 5, 2, 1, 3, 8, 1, 2, 4, 1, 3, 2, 1, 6, 5, 1, 2, 3, 1, 4, 2, 1, 7, 3, 1, 2, 11, 1, 4, 2, 1, 3, 5, 1, 2, 6, 1, 3, 2

Offset: 1

Enrico T. Federighi (rico125162(AT)aol.com), Aug 08 2003

If all the terms in the sequence are reduced by one and then all zeros are removed, the result is the same as the original sequence.
From Benoit Cloitre, Mar 07 2009: (Start)
To construct the sequence:
Step 1: start from a sequence of 1's and leave two undefined places between every pair of 1's giving: 1,(),(),1,(),(),1,(),(),1,(),(),1,(),(),1,...
Step 2: replace the first undefined place with a 2 and henceforth leave two undefined places between two 2's giving: 1,2,(),1,(),2,1,(),(),1,2,(),1,(),2,1,...
Step 3: replace the first undefined place with a 3 and henceforth leave two undefined places between two 3's giving: 1,2,3,1,(),2,1,(),3,1,2,(),1,(),2,1,...
Step 4: replace the first undefined place with a 4 and leave 2 undefined places between two 4's giving: 1,2,3,1,4,2,1,(),3,1,2,(),1,4,2,1,... Iterating the process indefinitely yields the sequence: 1,2,3,1,4,2,1,5,3,1,2,6,1,4,2,1,... (End)
From Peter Munn, Jul 10 2020: (Start)
For k >= 1, the number k occurs in a pattern with fundamental period 3^k, and with points of mirror symmetry at intervals of (3^k)/2. Those points have an extrapolated common origin (for k >= 1) at an offset 1.5 to the left of the sequence's initial "1". The snake format illustration in the example section may be useful for visualizing this.
(End)
For k >= 1, k first occurs at position A061419(k) and its k-th occurrence is at position A083045(k-1). - Peter Munn, Aug 23 2020
(a(n)) is the unique fixed point of the two-block substitution a,b -> 1,a+1,b+1, where a,b are natural numbers. - Michel Dekking, Sep 26 2022

			From _Peter Munn_, Jul 03 2020: (Start)
Listing the terms in a snake format (with period 27) illustrates periodic and mirror symmetries. Horizontal lines mark points of mirror symmetry for 3's. Vertical lines mark further points of mirror symmetry for 2's. 79 terms are shown. (Referred to the extrapolated common origin of periodic mirror symmetry, the initial term is at offset 1.5 and the last shown is at offset 79.5 = 3^4 - 1.5.) Observe also mirror symmetry of 4's (seen vertically).
    1  2  3  1  4  2  1  5   3  1  2  6
             |             |            1 --
    1  2  3  1  5  2  1  7   3  1  2  4
_ 4
  8
    1  2  3  1  6  2  1  4   3  1  2  5
             |             |            1 --
    1  2  3  1  7  2  1  4   3  1  2  9
_ 5
  4
    1  2  3  1  6  2  1 10   3  1  2  4
             |             |            1 --
    1  2  3  1  4  2  1  8   3  1  2  5
(End)
From _Peter Munn_, Aug 22 2020: (Start)
The start of the sequence is shown below in conjunction with related sequences, aligning their points of mirror symmetry. The longer, and shorter, vertical lines indicate points of mirror symmetry for terms valued less than 4, and less than 3, respectively. Note each term of A051064 is the minimum of two terms displayed nearest below it, and each term of A254046 is the minimum of the two terms displayed diagonally above it.
        |                          |                          |
A051064:| 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1 4 1 1 2
        |        |        |        |        |        |        |
[a(n)]: |  1 2 3 1 4 2 1 5 3 1 2 6 1 4 2 1 3 7 1 2 5 1 3 2 1 4 8 1 2 3
        |        |        |        |        |        |        |
A254046:|1 2 1 1 3 1 1 2 1 1 2 1 1 4 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1
        |                          |                          |
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A001511, A244040.
Sequences with equivalent symmetries: A051064, A254046.
Records are given by A061419: a(A061419(n))=n.
Essentially the odd bisection of A335933.
Sequence with similar definition: A087165.
Ordinal transform of A163491, with which this sequence has a joint relationship to A083044, A083045.
See also the comment in A024629.

A163491[n_] := A163491[n] = If[Mod[n, 3]==1, (n+2)/3, A163491[Floor[2n/3]]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A163491[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022 *)

a(n) = 1 when n == 1 (mod 3), otherwise a(n) = a(n-ceiling(n/3)) + 1.
a(n) = 3 + A244040(3*(n-1)) - A244040(3*n). - Tom Edgar and James Van Alstine, Aug 04 2014
From Peter Munn, Aug 22 2020: (Start)
For m >= 0, a(3*m+1) = 1; a(3*m+2) = a(2*m+1) + 1; a(3*m+3) = a(2*m+2) + 1.
For n >= 1, the following identities hold.
a(n) = A335933(2*n+1).
A083044(A163491(n) - 1, a(n) - 1) = n.
A051064(n+1) = min(a(n), a(n+1)).
A254046(n+2) = min(a(n), a(n+2)). (End)

More terms from Paul D. Hanna, Aug 21 2003
Offset changed by M. F. Hasler (following remarks by Peter Munn), Jul 13 2020
Thanks to Allan C. Wechsler for suggesting the new name. - N. J. A. Sloane, Jul 14 2020

A024631 n written in fractional base 4/3.

Original entry on oeis.org

0, 1, 2, 3, 30, 31, 32, 33, 320, 321, 322, 323, 3210, 3211, 3212, 3213, 32100, 32101, 32102, 32103, 32130, 32131, 32132, 32133, 321020, 321021, 321022, 321023, 321310, 321311, 321312, 321313, 3210200, 3210201, 3210202, 3210203, 3210230, 3210231

Offset: 0

David W. Wilson

G. C. Greubel, Table of n, a(n) for n = 0..1000
K. Burde, Das Problem der Abzählreime und Zahlentwicklungen mit gebrochenen Basen [The problem of counting rhymes and number expansions with fractional bases], J. Number Theory 26(2) (1987), 192-209. [The author deals with the representation of n in fractional bases k/(k-1) and its relation to counting-off games (variations of Josephus problem). Here k = 4. See the review in MathSciNet (MR0889384) by R. G. Stoneham.]
Index entries for sequences related to the Josephus Problem

Cf. A244041 (sum of digits).

Cf. A024629 (base 3/2).

a:= proc(n) `if`(n<1, 0, irem(n, 4, 'q')+a(3*q)*10) end:
seq(a(n), n=0..45);  # Alois P. Heinz, Aug 20 2019

p:= 4; q:= 3; a[n_]:= a[n]= If[n==0, 0, 10*a[q*Floor[n/p]] + Mod[n, p]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Aug 20 2019 *)

a(n) = my(p=4, q=3); if(n==0,0, 10*a(q*(n\p)) + (n%p));
vector(40, n, n--; a(n)) \\ G. C. Greubel, Aug 20 2019

def basepqExpansion(p, q, n):
    L, i = [n], 1
    while L[i-1] >= p:
        x=L[i-1]
        L[i-1]=x.mod(p)
        L.append(q*(x//p))
        i+=1
    return Integer(''.join(str(x) for x in reversed(L)))
[basepqExpansion(4,3,n) for n in [0..40]] # G. C. Greubel, Aug 20 2019

To represent a number in base b, if a digit is greater than or equal to b, subtract b and carry 1. In fractional base a/b, subtract a and carry b.

A303500 The smallest positive even integer that can be written with n digits in base 3/2.

Original entry on oeis.org

2, 21, 210, 2101, 21011, 210110, 2101100, 21011000, 210110001, 2101100011, 21011000110, 210110001101, 2101100011010, 21011000110100, 210110001101001, 2101100011010011, 21011000110100110, 210110001101001101

Offset: 0

Tanya Khovanova and PRIMES STEP Senior group, May 09 2018

a(n) is a prefix of a(n+1).

The smallest, not necessarily even, integer in base 3/2 with n digits is a(n-1) with 0 added at the end.

			The number 5 in base 3/2 is 22, and the number 6 is 210. Therefore, 210 is the smallest even integer with 3 digits in base 3/2.

B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.

See A024629 for the base-3/2 expansion of n.

Cf. also A304024, A304025, A070885, A304272, A081848, A246435, A005428, A073941.

roll32 := proc(L)
    local piv,L1 ;
    piv := 1;
    L1 := subsop(piv=op(piv,L)+1,L) ;
    while op(piv,L1) >= 3 do
        L1 := [seq(0,i=1..piv), op(piv+1,L1)+1, seq(op(i,L1),i=piv+2..nops(L1))] ;
        piv := piv+1 ;
    end do:
    L1 ;
end proc:
from32 := proc(L)
    add( op(i,L)*(3/2)^(i-1),i=1..nops(L)) ;
end proc:
A303500 := proc(n)
    local dgs ;
    dgs := [seq(0,i=1..n-1),1] ;
    while not type(from32(dgs),'even') do
        dgs := roll32(dgs) ;
    end do:
    dgs := ListTools[Reverse](dgs) ;
    digcatL(%) ;
end proc: # R. J. Mathar, Jun 25 2018

a(n) = A024629(A305498(n)). - R. J. Mathar, Jun 25 2018

cp's OEIS Frontend

A081848 Number of numbers whose base-3/2 expansion (see A024629) has n digits.

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A304273 The concatenation of the first n terms is the smallest positive even number with n digits when written in base 3/2 (cf. A024629).

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A005428 a(n) = ceiling((1 + sum of preceding terms) / 2) starting with a(0) = 1.

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A244040 Sum of digits of n in fractional base 3/2.

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A342426 Niven numbers in base 3/2: numbers divisible by their sum of digits in fractional base 3/2 (A244040).

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A246435 Length of representation of n in fractional base 3/2.

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A070885 a(n) = (3/2)*a(n-1) if a(n-1) is even; (3/2)*(a(n-1)+1) if a(n-1) is odd.

A070885 a(n) = (3/2)a(n-1) if a(n-1) is even; (3/2)(a(n-1)+1) if a(n-1) is odd.