A106370 -id:A106370 - cp's OEIS frontend

A106371 Representation of n in base b, where b is minimal such that n contains no zeros: b = A106370(n).

1, 2, 11, 11, 12, 12, 111, 22, 21, 22, 23, 22, 111, 112, 1111, 121, 122, 33, 34, 32, 111, 211, 212, 44, 221, 222, 123, 44, 131, 132, 11111, 112, 113, 114, 55, 121, 211, 212, 213, 1111, 1112, 222, 1121, 1122, 231, 232, 233, 143, 1211, 1212, 123, 1221, 1222, 312

Offset: 1

Reinhard Zumkeller, May 01 2005

In decimal representation feasible only for bases <= 10; A106371(11) = 360 = 2*11^2 + 10*11^1 + 8*11^0 is the first number which cannot be written zerofree with digits 1..9 in base 11. Therefore this sequence is finite with exactly 359 terms. - Reinhard Zumkeller, Apr 12 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..359

Cf. A106370, A004086.

a106371 n = a106371_list !! (n-1)
a106371_list = map fromJust $ takeWhile (/= Nothing) $ map f [1..] where
   f n = g 2 n where
     g b x = h x 0 where
       h 0 y = if b <= 10 then Just (a004086 y) else Nothing
       h z y = if r == 0 then g (b + 1) n else h z' (10 * y + r)
               where (z', r) = divMod z b
-- Reinhard Zumkeller, Apr 12 2015

A119352 Smallest base b > 1 such that n in base b uses no digit b-1.

Original entry on oeis.org

2, 3, 4, 3, 3, 4, 4, 5, 4, 3, 3, 5, 3, 3, 6, 5, 4, 4, 4, 6, 4, 4, 4, 7, 4, 4, 4, 3, 3, 7, 3, 3, 4, 4, 4, 5, 3, 3, 4, 3, 3, 4, 4, 5, 6, 6, 6, 9, 6, 6, 5, 5, 5, 5, 6, 5, 5, 5, 5, 7, 5, 5, 5, 5, 4, 4, 4, 5, 4, 4, 4, 7, 4, 4, 4, 5, 5, 5, 5, 6, 4, 3, 3, 5, 3, 3, 4, 5, 4, 4, 3, 3, 5, 3, 3, 9, 4, 4, 4, 6, 4, 4, 4, 7, 4

Offset: 0

Franklin T. Adams-Watters, May 15 2006

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A106370, A119353, A119354.

a119352 n = f 2 n where
   f b x = g x where
     g 0 = b
     g z = if r == b - 1 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], b-1], b++]; b]; Array[a, 100, 0] (* Amiram Eldar, Jul 29 2025 *)

a(n) = {my(b = 2, d); while(d = digits(n, b); #d > 0 && vecmax(d) == b-1, b++); b;} \\ Amiram Eldar, Jul 29 2025

7 is 111_2 (has 1), 21_3 (has 2), 13_4 (has 3), but 12_5 (no 4), so a(7) = 5.

A106372 Smallest number m containing no zeros in base n representation but at least one zero in all base b representations with 1

Original entry on oeis.org

1, 2, 6, 12, 20, 60, 147, 384, 896, 360, 1540, 2760, 2304, 11016, 13860, 6720, 123060, 255024, 658370, 1422720, 1425606, 1179360, 2920500, 12252240, 9989280, 24227280, 242573760, 68424048, 990208128, 410810400, 602751600

Offset: 2

Reinhard Zumkeller, May 01 2005

A106370(a(n)) = n and A106370(m) <> n for m < a(n).

a(46) = 854710136280. a(45) and all further terms are at least 10^13.

			a(10)=896, A106370(896)=10: 896[base-9]='1205',
896[octal]='1600', 896[base-7]='2420', 896[base-6]='4052',
896[base-5]='12041', 896[base-4]='32000', 896[ternary]='1020012',
896[binary]='1110000000'.

Michael Gottlieb, Table of n, a(n) for n = 2..44 (terms 2..32 from Reinhard Zumkeller, terms 33..44 from Michael Gottlieb).
Michael Gottlieb, C++ program to compute values.

Cf. A106370, A119354.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a106372 = (+ 1) . fromJust . (`elemIndex` a106370_list)
-- Reinhard Zumkeller, Apr 12 2015

Generated a(33)-a(44) and a(46), with linked C++ program

cp's OEIS Frontend

A106371 Representation of n in base b, where b is minimal such that n contains no zeros: b = A106370(n).

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A119352 Smallest base b > 1 such that n in base b uses no digit b-1.

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A106372 Smallest number m containing no zeros in base n representation but at least one zero in all base b representations with 1

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