A388299 a(n) is the least base b >= 2 such that the base-b representation of n is the base-10 representation of a prime.

3, 2, 3, 2, 5, 4, 5, 6, 3, 4, 9, 4, 11, 6, 5, 4, 11, 4, 11, 5, 3, 2, 15, 6, 5, 6, 5, 4, 21, 7, 23, 5, 3, 8, 27, 4, 7, 6, 11, 5, 33, 4, 35, 6, 3, 2, 9, 8, 7, 11, 7, 4, 45, 4, 5, 8, 3, 7, 51, 4, 53, 5, 9, 8, 9, 3, 5, 12, 21, 4, 63, 4, 13, 8, 23, 4, 11, 3, 11, 5, 9, 4, 9, 7, 11, 6, 9, 2, 81, 4, 13

Offset

2,1

Comments

The base-b representation of n must not use any digits > 9.

For n >= 19, a(n) <= n-9 since the base-(n-9) representation of n is 19.

If n >= 29 is odd, a(n) <= (n-9)/2 since the base-((n-9)/2) representation of n is 29.

Links

Robert Israel, Table of n, a(n) for n = 2..10000

Formula

A386929(n) = a(n) if a(n) <= 10, otherwise A386929(n) = 0.

Example

a(7) = 4 because 7 = 13_4, 13 is prime, and no smaller number than 4 works.
a(14) = 11 because 14 = 13_11,

Maple

f:= proc(n) local b, L, i, p;
  for b from 2 do
    L:= convert(n, base, b);
    if max(L) > 9 then next fi;
    p:= add(L[i]*10^(i-1), i=1..nops(L));
    if isprime(p) then return b fi
  od
end proc:
map(f, [$2..100]);

Mathematica

a[n_]:=Module[{b=2}, While[!ContainsOnly[IntegerDigits[n, b], Range[0, 9]]||!PrimeQ[FromDigits[IntegerDigits[n, b]]], b++]; b]; Array[a, 91, 2] (* James C. McMahon, Sep 22 2025 *)

Prog

(PARI) isok(n, b) = my(d=digits(n, b)); (vecmax(d) <=9) && isprime(fromdigits(d));
a(n) = my(b=2); while(!isok(n, b), b++); b; \\ Michel Marcus, Sep 16 2025

Crossrefs

Cf. A036952, A052026, A386929, A052033, A388300.

Sequence in context: A112924 A267148 A386929 * A367682 A230406 A362688

Adjacent sequences: A388296 A388297 A388298 * A388300 A388301 A388302

Keyword

nonn,base,look

Author

Robert Israel, Sep 16 2025

Status

approved