A106370 Smallest b > 1 such that n contains no zeros in its base b representation.

2, 3, 2, 3, 3, 4, 2, 3, 4, 4, 4, 5, 3, 3, 2, 3, 3, 5, 5, 6, 4, 3, 3, 5, 3, 3, 4, 6, 4, 4, 2, 5, 5, 5, 6, 5, 4, 4, 4, 3, 3, 4, 3, 3, 4, 4, 4, 5, 3, 3, 6, 3, 3, 4, 4, 5, 4, 4, 4, 7, 4, 4, 2, 5, 6, 5, 3, 3, 5, 3, 3, 5, 5, 5, 7, 3, 3, 7, 3, 3, 5, 5, 5, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 5, 5, 5, 6, 4, 4, 4, 6, 4

Offset: 1

Author

Reinhard Zumkeller, May 01 2005

Keywords

• nonn
• base

Examples

			n = 20: 20[binary] = '101001', 20[ternary] = '202', 20[base-4] = '110', 20[base-5] = '40', all containing at least one zero, but: 20[base-6] = '32', containing no zero therefore a(20) = 6.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A106371.

Cf. A000225, A032924, A106372, A119352.

Programs

Haskell

a106370 n = f 2 n where
   f b x = g x where
     g 0 = b
     g z = if r == 0 then f (b + 1) n else g z'
           where (z', r) = divMod z b
-- Reinhard Zumkeller, Apr 12 2015

Mathematica

a[n_] := Module[{b = 2}, While[MemberQ[IntegerDigits[n, b], 0], b++]; b]; Array[a, 100] (* Amiram Eldar, Jul 29 2025 *)

Formula

a(n*a(n)+k) <= a(n) for 1 <= k < a(n).

a(A106372(n)) = n and a(m) <> n for m < A106372(n).

a(A000225(n)) = 2; a(A032924(n)) = 3 for n <> 5.

Extensions

Typo in comment fixed by Reinhard Zumkeller, Aug 06 2010