A119352 -id:A119352 - cp's OEIS frontend

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

Reinhard Zumkeller, May 01 2005

			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.

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

Cf. A106371.

Cf. A000225, A032924, A106372, A119352.

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

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

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.

Typo in comment fixed by Reinhard Zumkeller, Aug 06 2010

A119354 Smallest number which does not use digit n-1 written in base n, but does use digit b-1 written in base b for any 1

Original entry on oeis.org

0, 1, 2, 7, 14, 23, 139, 47, 485, 319, 989, 239, 5927, 23319, 27719, 13679, 57119, 159119, 746927, 500399, 1627919, 2585519, 3694397, 7305479, 17067599, 82265039, 384812999, 41126399, 61483967, 586370399, 589188599, 2196179999, 1770310079

Offset: 2

Franklin T. Adams-Watters, May 15 2006

Cf. A106372, A119352.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a119354 = fromJust . (`elemIndex` a119352_list)
-- Reinhard Zumkeller, Apr 12 2015

A119352(a(n)) = n and A119352(m) != n for m < a(n). - Reinhard Zumkeller, Apr 12 2015

More terms from R. J. Mathar, Sep 07 2007
a(21)-a(27) by Reinhard Zumkeller, Apr 12 2015
a(28)-a(34) from Amiram Eldar, Jul 29 2025

A119353 Smallest value obtained by writing n in base b > 1 such that n in base b uses no digit b-1 and then interpreting this as a base b-1 number.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 5, 6, 6, 4, 5, 9, 6, 7, 12, 12, 9, 10, 11, 16, 12, 13, 14, 20, 15, 16, 17, 8, 9, 25, 10, 11, 18, 19, 20, 24, 12, 13, 23, 14, 15, 25, 26, 31, 32, 33, 34, 42, 35, 36, 32, 33, 34, 35, 40, 36, 37, 38, 39, 45, 40, 41, 42, 43, 27, 28, 29, 46, 30, 31, 32, 55, 33, 34, 35

Offset: 0

Franklin T. Adams-Watters, May 15 2006

The b that produces the smallest value is always the smallest b such that n in base b does not use b-1.

			10 = 101_3, 101_2 = 5, so a(10) = 5.
11 = 102_3, which has a 2; 11 = 23_4, which has a 3; 11 = 21_5, 21_4 = 9, so a(11) = 9.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A119352.

a[n_] := Module[{b = 2}, While[MemberQ[IntegerDigits[n, b], b-1], b++]; FromDigits[IntegerDigits[n, b], b-1]]; Array[a, 100, 0] (* Amiram Eldar, Jul 29 2025 *)

a(n) = for (b=3, oo, my (d=digits(n, b)); if (#d==0 || vecmax(d) < b-1, return (fromdigits(d, b-1)))) \\ Rémy Sigrist, May 07 2018

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

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A119354 Smallest number which does not use digit n-1 written in base n, but does use digit b-1 written in base b for any 1

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A119353 Smallest value obtained by writing n in base b > 1 such that n in base b uses no digit b-1 and then interpreting this as a base b-1 number.

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Mathematica

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