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

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

Offset: 1

Author

Tanya Khovanova and PRIMES STEP Senior group, May 04 2018

Keywords

• nonn
• base

Comments

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.

Examples

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

Links

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

Crossrefs

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

Programs

Maple

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

PARI

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

Python

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

Formula

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

Extensions

More terms from Michel Marcus, Jun 19 2018