A216194 a(n) = Smallest b for which the base b representation of n contains at least one 2 (or 0 if no such base exists).

0, 3, 0, 0, 3, 3, 3, 3, 4, 4, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 10, 3, 4, 11, 3, 3, 3, 3, 4, 4, 3, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 6, 4, 3, 6, 5, 3, 3, 3, 3, 4, 4, 3, 6, 4, 3, 3, 3, 3, 3, 3

Offset: 1

Author

Nathan Fox, Mar 12 2013

Keywords

• nonn
• easy
• base

Comments

a(n)=3 if and only if n is in A074940.

a(n) > 0 for n >= 5 since 12 is n written in base n-2.

The only perfect k-th powers (k>=2) that can appear in this sequence are 2^k with k a prime number.

The first n for which a(n)=7 is 849.

The first n for which a(n)=8 is 1084.

The first n for which a(n)=10 is 28. The second is 243.

The first n for which a(n)=11 is 31. The second is 58130496.

a(n)<=11 for all n with fewer than 3000 base 10 digits. No n for which a(n)>11 has been found.

Links

• Nathan Fox, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A216192, A270027, A270028, A270029, A270030, A270031, A270032, A270033, A270034, A270035.

Programs

Maple

firstNTerms:=proc(n) local b,i,rep,L:
L:=[]:
for i from 5 to n do
    b:=3:
    while true do
        rep:=convert(i, base, b):
        if evalb(2 in rep) then
            L:=[op(L), b]:
            break:
        fi:
        b:=b+1:
    od:
od:
L:
end:

Mathematica

sb2[n_]:=Module[{b=3},While[DigitCount[n,b,2]<1,b++];b]; Array[sb2,110,5] (* Harvey P. Dale, Jan 16 2016 *)
Table[SelectFirst[Range[3, 1200], DigitCount[n, #, 2] > 0 &], {n, 5, 120}] (* Michael De Vlieger, Mar 09 2016, Version 10 *)

PARI

a(n) = if ((n<5) && (n!=2), 0, my(b=3); while (! vecsearch(vecsort(digits(n, b)), 2), b++); b);  \\ Michel Marcus, Aug 06 2014, Mar 11 2016

Extensions

Modified the definition to make the offset 1 by Nathan Fox, Mar 10 2016