A364052 a(n) is the least k such that no number with distinct base-n digits is the product of k (not necessarily distinct) primes.

2, 3, 7, 9, 12, 14, 14, 21, 26, 28, 33, 36, 40, 45, 36, 50, 59, 61, 65, 70, 75, 77, 85, 89, 94, 97, 104, 107, 113, 118, 84

Offset: 2

Author

Robert Israel, Jul 04 2023

Keywords

• nonn
• base
• hard
• more

Comments

A364049(n) <= a(n) <= 1 + floor(log_2(A062813(n))).

Examples

			a(4) = 7 because 2 = 2^1 = 2_4, 4 = 2^2 = 10_4, 8 = 2^3 = 20_4, 24 = 2^3 * 3 = 120_4, 108 = 2^2 * 3^3 = 1230_4 and 216 = 2^3 * 3^3 = 3120_4 have distinct base-4 digits and are products of 1 to 6 primes respectively, but there is no product of 7 primes that has distinct base-4 digits.
		

Crossrefs

Cf. A001222, A062813, A165712, A364049.

Programs

Maple

f:= proc(n) local d,S,V,k;
  V:= {};
  for d from 1 to n do
    S:= select(t -> t[-1] <> 0, combinat:-permute([$0..n-1],d));
    S:= map(proc(t) local i;  numtheory:-bigomega(add(t[i]*n^(i-1),i=1..d)) end proc, S);
    V:= V union convert(S,set);
  od;
  min({$1..1+max(V)} minus V)
end proc:
map(f, [$2..10]);

Extensions

a(11) from Jon E. Schoenfield, Jul 05 2023

a(12) from Martin Ehrenstein, Jul 07 2023

a(13)-a(18) from Jon E. Schoenfield, Jul 08 2023

a(19)-a(22) from Pontus von Brömssen, Jul 13 2023

a(23)-a(32) from Bert Dobbelaere, Jul 20 2023