A304273 -id:A304273 - cp's OEIS frontend

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

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

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

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Author

Keywords

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Programs

Python

Formula