Voozh

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

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OFFSET

1,1

LINKS

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

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.

EXAMPLE

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.

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A106371.