A304025 -id:A304025 - cp's OEIS frontend

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

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

A304024 a(n) is the largest integer with n digits in base 3/2.

Original entry on oeis.org

2, 22, 212, 2122, 21222, 212212, 2122112, 21221112, 212211122, 2122111222, 21221112212, 212211122122, 2122111221212, 21221112212112, 212211122121122, 2122111221211222, 21221112212112212, 212211122121122122

Offset: 0

Tanya Khovanova and PRIMES STEP Senior group, May 04 2018

Every number starts and ends with 2 and contains only twos and ones.
Removing the last digit produces sequence A304272 of the largest even integers in base 3/2.
The value of this sequence in base 10 is A304025.
When adding 1 to the value of this sequence we get A070885.
The largest integer with a given number of digits in base 3/2 can be produced directly from the smallest number, sequence A304023, by replacing 21 at the beginning and 0 at the end with 2, and by shifting the rest up by 1, see sequence A304023.

			The number 5 in base 3/2 is 22, and the number 6 is 210. Therefore, 22 is the largest two-digit integer.

Cf. A005428, A024629, A070885, A073941, A081848, A246435, A303500, A304023, A304025, A304272.

first(n) = {my(res=vector(n), c = 2); res[1]=2; for(i=2, n, res[i] = 10 * res[i-1] + 2; if(c % 2 == 1, res[i] -= 10); c = 3 * c / 2 + if(c%2==0, 2, 1/2)); res} \\ David A. Corneth, May 11 2018

a(1) = 2, for n > 1, a(n) = 10 * a(n - 1) + 2 if A304025(n - 1) is even. Otherwise, a(n) = 10 * a(n - 1) - 8. - David A. Corneth, May 11 2018

A304272 The largest even integer that can be written with n digits in base 3/2.

Original entry on oeis.org

2, 21, 212, 2122, 21221, 212211, 2122111, 21221112, 212211122, 2122111221, 21221112212, 212211122121, 2122111221211, 21221112212112, 212211122121122, 2122111221211221, 21221112212112212, 212211122121122121, 2122111221211221212, 21221112212112212121, 212211122121122121211, 2122111221211221212112

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 09 2018

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

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

			The number 4 in base 3/2 is 21, and number 6 is 210. Therefore, 21 is the largest even integer with 2 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.

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

Table[StringTake["212211122121122121211221211212112", n], {n, 32}]

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.

A304023 a(n) is the smallest integer with n digits in base 3/2 expressed in base 3/2.

Original entry on oeis.org

0, 20, 210, 2100, 21010, 210110, 2101100, 21011000, 210110000, 2101100010, 21011000110, 210110001100, 2101100011010, 21011000110100, 210110001101000, 2101100011010010, 21011000110100110, 210110001101001100, 2101100011010011010, 21011000110100110100, 210110001101001101010

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 04 2018

Excluding 0, every term starts with 2 and has exactly one 2.
The last digit is always zero.
Removing the last digit produces the sequence A303500 of the smallest even integers in base 3/2.
The value of this sequence in base 10 is A070885.
When subtracting 1 from the value of this sequence we get A304025.
The largest integer with a given number of digits in base 3/2 can be produced directly from this sequence by replacing 21 at the beginning and 0 at the end with 2, and by shifting the rest up by 1, see sequence A304024.

			The number 5 in base 3/2 is 22, and the number 6 is 210. Therefore, 210 is the smallest three-digit integer.

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

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

b:= proc(n) b(n):= `if`(n=1, 1, 3*ceil(b(n-1)/2)) end:
g:= proc(n) g(n):= `if`(n<2, 0, irem(n, 3, 'q')+g(2*q)*10) end:
a:= n-> g(b(n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 13 2021

f(n) = if( n<1, 0, f(n\3 * 2) * 10 + n%3);
a(n) = {my(k=0); while(#Str(f(k)) != n, k++); f(k);} \\ Michel Marcus, Jun 19 2018

def f(n): return 0 if n < 1 else f(n//3*2)*10 + n%3
def a(n):
  k = 0
  while len(str(f(k))) != n: k += 1
  return f(k)
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Feb 12 2021 after Michel Marcus

a(n) = A024629(A070885(n)). - Michel Marcus, Jun 19 2018

More terms from Michel Marcus, Jun 19 2018

A304274 The concatenation of the first n elements is the largest positive even number with n digits when written in base 3/2.

Original entry on oeis.org

2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 09 2018

This sequence is possible due to the fact that the largest even integers are prefixes of each other.

A304272(n) is the largest even integer with n digits.

			Number 8 in base 3/2 is 212, and it is the largest even integer with 3 digits in base 3/2. Its prefix 21 is 4: the largest even integer with 2 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.

Cf. A005428, A070885, A073941, A081848, A024629, A246435, A304024, A304025, A303500, A304272, A304273, A305497, A305498.

b:= proc(n) option remember; `if`(n=1, 2,
      (t-> t+irem(t, 2))(b(n-1)*3/2))
    end:
a:= n-> b(n+1)-3/2*b(n)+1:
seq(a(n), n=1..120);  # Alois P. Heinz, Jun 21 2018

b[n_] := b[n] = If[n == 1, 2, Function[t, t + Mod[t, 2]][3/2 b[n-1]]];
a[n_] := b[n+1] - 3/2 b[n] + 1;
Array[a, 120] (* Jean-François Alcover, Dec 13 2018, after Alois P. Heinz *)

a(n) = A304273(n+1) + 1.
From Alois P. Heinz, Jun 21 2018: (Start)
a(n) = A305498(n+1) -3/2*A305498(n) + 1.
Sum_{i=0..n-1} (3/2)^i*a(n-i) = A305497(n). (End)

More terms from Alois P. Heinz, Jun 21 2018

A305497 The largest positive even integer that can be represented with n digits in base 3/2.

Original entry on oeis.org

2, 4, 8, 14, 22, 34, 52, 80, 122, 184, 278, 418, 628, 944, 1418, 2128, 3194, 4792, 7190, 10786, 16180, 24272, 36410, 54616, 81926, 122890, 184336, 276506, 414760, 622142, 933214, 1399822, 2099734, 3149602, 4724404, 7086608, 10629914, 15944872, 23917310

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, Jun 02 2018

Michael De Vlieger, 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.

Cf. A005428, A070885, A073941, A081848, A024629, A246435, A304023, A304024, A304025, A303500, A304272, A304273, A304274, A305498.

b[n_] := b[n] = If[n == 1, 2, Function[t, t + Mod[t, 2]][3/2 b[n - 1]]]; a[n_] := b[n + 1] - 3/2 b[n] + 1; A305497[n_] := Sum[(3/2)^i*a[n - i], {i, 0, n - 1}]; Table[A305497[n], {n, 1, 39}] (* Robert P. P. McKone, Feb 12 2021 *)

from itertools import islice
def A305497_gen(): # generator of terms
    a = 2
    while True:
        a += a>>1
        yield (a<<1)-4
A305497_list = list(islice(A305497_gen(),70)) # Chai Wah Wu, Sep 20 2022

a(n+1) = 2*floor(3*a(n)/4) + 2.
a(n) = 2*A061419(n+1) - 2.
a(n) = A305498(n+1) - 2.
a(n) = Sum_{i=0..n-1} (3/2)^i*A304274(n-i). - Alois P. Heinz, Jun 21 2018

A305498 The smallest positive even integer that can be represented with n digits in base 3/2.

Original entry on oeis.org

2, 4, 6, 10, 16, 24, 36, 54, 82, 124, 186, 280, 420, 630, 946, 1420, 2130, 3196, 4794, 7192, 10788, 16182, 24274, 36412, 54618, 81928, 122892, 184338, 276508, 414762, 622144, 933216, 1399824, 2099736, 3149604, 4724406, 7086610, 10629916, 15944874, 23917312

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, Jun 02 2018

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, A304023, A304024, A304025, A303500, A304272, A304273, A304274, A305497.

from itertools import islice
def A305498_gen(): # generator of terms
    a = 2
    while True:
        yield (a<<1)-2
        a += a>>1
A305498_list = list(islice(A305498_gen(),70)) # Chai Wah Wu, Sep 20 2022

a(n+1) = 2*ceiling(3*a(n)/4).
a(n) = 2*A061419(n).
a(n) = A305497(n-1) + 2.

cp's OEIS Frontend

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

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A304024 a(n) is the largest integer with n digits in base 3/2.

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A304272 The largest even integer that can be written with n digits in base 3/2.

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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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A304023 a(n) is the smallest integer with n digits in base 3/2 expressed in base 3/2.

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A304274 The concatenation of the first n elements is the largest positive even number with n digits when written in base 3/2.

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A305497 The largest positive even integer that can be represented with n digits in base 3/2.

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