A119354 -id:A119354 - cp's OEIS frontend

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

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

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

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

f:= proc(n) local b;
    for b from 2 do if not member(b-1,convert(n,base,b)) then return b fi od
end proc:
map(f, [$0..200]); # Robert Israel, Sep 07 2025

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

from itertools import count
from sympy.ntheory import digits
def a(n): return next(b for b in count(2) if b-1 not in digits(n, b))
print([a(n) for n in range(105)]) # Michael S. Branicky, Sep 07 2025

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

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

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