A362830 Number of bases b with 2 <= b < n such that n written in base b is a strictly increasing sequence of digits.

0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 4, 6, 6, 7, 7, 9, 8, 11, 10, 11, 12, 14, 12, 15, 15, 17, 16, 19, 17, 20, 19, 21, 22, 23, 21, 25, 26, 27, 25, 29, 27, 30, 30, 30, 32, 34, 31, 35, 34, 37, 37, 40, 37, 40, 39, 41, 43, 45, 40, 46, 46, 46, 46, 49, 48, 52, 51, 54, 51

Offset: 1

Author

Matthew R. Maas, May 04 2023

Keywords

• nonn
• base

Examples

			The number 27 forms a strictly increasing sequence of digits when written in base 4 (1,2,3), base 7 (3,6), base 10 (2,7), base 11 (2,5), base 12 (2,3), and bases 14 through 25 (1,13 through 1,2), and no other bases below 27. There are 17 bases with this property, so a(27)=17.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Programs

Mathematica

a[n_] := Count[Range[2, n], ?(Less @@ IntegerDigits[n, #] &)]; Array[a, 100] (* _Amiram Eldar, Aug 02 2023 *)

PARI

a(n) = sum(b=2, n-1, my(d=digits(n,b)); d==Set(d)); \\ Michel Marcus, May 07 2023

Python

from sympy.ntheory import digits
def c(v): return all(v[i+1] > v[i] for i in range(len(v)-1))
def a(n): return sum(1 for b in range(2, n) if c(digits(n, b)[1:]))
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, May 05 2023

Extensions

a(33) and beyond from Michael S. Branicky, May 05 2023