A326775 For any n >= 0, let b >= 2 be the smallest base where n has all digits equal, say to d; a(n) = d.

0, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 5, 4, 1, 2, 3, 1, 1, 2, 1, 4, 5, 2, 1, 6, 1, 5, 3, 4, 1, 6, 5, 4, 1, 2, 1, 6, 1, 2, 1, 4, 5, 6, 1, 4, 3, 7, 1, 6, 1, 2, 5, 4, 7, 6, 1, 2, 3, 2, 1, 7, 1, 2

Offset: 0

Author

Rémy Sigrist, Jul 28 2019

Keywords

• nonn
• base

Comments

A059711 gives base b.

From Bernard Schott, Aug 17 2019: (Start)

a(n) = 1 iff n is A220570, then n = 11_(n-1) or, n is in A053696, then n = 11..11_b for some base b.

a(n) = 2 if n = 2 * p, p prime >= 5.

a(n) = 3 if n = 3 * p, p prime >= 11.

There are k = 2 equal digits in the representation of n in the corresponding base b, except when n is a term of A167782, in which case the number k of equal digits is >= 3. (End)

n = (b^k - 1)/(b - 1) * a(n) so a(n) | n for n > 0. Furthermore a(n) <= sqrt(n). - David A. Corneth, Aug 21 2019

If b is the smallest base such that n=d*b^k+...+d*b^0 (A059711) (d=a(n) is the repdigit) then n mod b = (d*b^k+...+d*b^0) mod b = (d*b^k+...+d*b^1) mod b + (d*b^0) mod b = 0 + (d*1) mod b. Since d is less than the base we end up with the formula n mod b = d. - Jon Maiga, May 31 2021

Examples

			For n = 45:
- we have:
     b  45 in base b  Repdigit ?
     -  ------------  ----------
     2  101101        no
     3  1200          no
     4  231           no
     5  140           no
     6  113           no
     7  63            no
     8  55            yes, with d = 5
- hence a(45) = 5.
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..9999

Crossrefs

Cf. A059711, A053696, A060775, A220570.

Programs

PARI

a(n) = for (b=2, oo, if (#Set(digits(n,b))<=1, return (n%b)))

Python

# with library / without (faster for large n)
from sympy.ntheory import digits
def is_repdigit(n, b): return len(set(digits(n, b)[1:])) == 1
def is_repdigit(n, b):
  if n < b: return True
  n, r = divmod(n, b)
  onlyd = r
  while n > b:
    n, r = divmod(n, b)
    if r != onlyd: return False
  return n == onlyd
def a(n):
  for b in range(2, n+3):
    if is_repdigit(n, b): return n%b
print([a(n) for n in range(87)]) # Michael S. Branicky, May 31 2021

Formula

n is a multiple of a(n).

a(n) = n mod A059711(n). - Jon Maiga, May 31 2021