A390088 a(n) is the least integer base b > 1 in which the representation of n uses only prime digits; or 0 if no such base exists.

0, 0, 3, 4, 0, 6, 0, 8, 3, 0, 4, 4, 5, 5, 4, 4, 7, 5, 5, 7, 6, 6, 10, 6, 7, 9, 3, 8, 13, 8, 9, 8, 6, 6, 9, 6, 11, 7, 7, 12, 7, 12, 4, 4, 13, 8, 4, 4, 9, 14, 9, 16, 9, 10, 17, 10, 15, 10, 4, 4, 11, 8, 4, 4, 17, 9, 9, 5, 5, 22, 9, 12, 10, 10, 19, 10, 13, 10, 15, 11

Offset

0,3

Comments

a(p) > 0 for any prime p because p is the only digit in base b = p + 1.

Conjecture: Exactly the values k = 0, 1, 4, 6, 9 satisfy a(k) = 0.

Links

Table of n, a(n) for n=0..79.

Example

a(11) = 4 because 11 is (2)(3) in base 4 where 2 and 3 are primes and there is no smaller base with that property.
a(74) = 19 because 74 is (3)(17) in base 19 where 3 and 17 are primes and there is no smaller base with that property.

Maple

A390088:=proc(n)
    local b, k;
    for b from 2 to n+1 do
        k:=convert(n, 'base', b);
        if select(isprime, k)=k then
            return b
        fi
    od;
    return 0
end proc;
seq(A390088(n), n=0..79);

Mathematica

a[n_]:=Module[{b=2}, If[MemberQ[{0, 1, 4, 6, 9}, n], 0, b=2; While[!AllTrue[IntegerDigits[n, b], PrimeQ], b++]; b]]; Array[a, 80, 0] (* James C. McMahon, Oct 31 2025 *)

Prog

(PARI) a(n) = for(b=2, n+1, my(d=digits(n, b)); if (#select(isprime, d) == #d, return(b))); \\ Michel Marcus, Oct 29 2025

Crossrefs

Cf. A355034, A384211.

Sequence in context: A383007 A158677 A337164 * A105576 A105826 A110665

Adjacent sequences: A390085 A390086 A390087 * A390089 A390090 A390091

Keyword

nonn,base,easy

Author

Felix Huber, Oct 29 2025

Status

approved