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

%I A106370 #14 Jul 30 2025 09:46:26
%S A106370 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,
%T A106370 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,
%U A106370 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
%N A106370 Smallest b > 1 such that n contains no zeros in its base b representation.
%H A106370 Reinhard Zumkeller, <a href="/A106370/b106370.txt">Table of n, a(n) for n = 1..10000</a>
%F A106370 a(n*a(n)+k) <= a(n) for 1 <= k < a(n).
%F A106370 a(A106372(n)) = n and a(m) <> n for m < A106372(n).
%F A106370 a(A000225(n)) = 2; a(A032924(n)) = 3 for n <> 5.
%e A106370 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.
%t A106370 a[n_] := Module[{b = 2}, While[MemberQ[IntegerDigits[n, b], 0], b++]; b]; Array[a, 100] (* _Amiram Eldar_, Jul 29 2025 *)
%o A106370 (Haskell)
%o A106370 a106370 n = f 2 n where
%o A106370    f b x = g x where
%o A106370      g 0 = b
%o A106370      g z = if r == 0 then f (b + 1) n else g z'
%o A106370            where (z', r) = divMod z b
%o A106370 -- _Reinhard Zumkeller_, Apr 12 2015
%Y A106370 Cf. A106371.
%Y A106370 Cf. A000225, A032924, A106372, A119352.
%K A106370 nonn,base
%O A106370 1,1
%A A106370 _Reinhard Zumkeller_, May 01 2005
%E A106370 Typo in comment fixed by _Reinhard Zumkeller_, Aug 06 2010