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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.

%I A364052 #33 Jul 20 2023 12:12:33
%S A364052 2,3,7,9,12,14,14,21,26,28,33,36,40,45,36,50,59,61,65,70,75,77,85,89,
%T A364052 94,97,104,107,113,118,84
%N 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.
%C A364052 A364049(n) <= a(n) <= 1 + floor(log_2(A062813(n))).
%e A364052 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.
%p A364052 f:= proc(n) local d,S,V,k;
%p A364052   V:= {};
%p A364052   for d from 1 to n do
%p A364052     S:= select(t -> t[-1] <> 0, combinat:-permute([$0..n-1],d));
%p A364052     S:= map(proc(t) local i;  numtheory:-bigomega(add(t[i]*n^(i-1),i=1..d)) end proc, S);
%p A364052     V:= V union convert(S,set);
%p A364052   od;
%p A364052   min({$1..1+max(V)} minus V)
%p A364052 end proc:
%p A364052 map(f, [$2..10]);
%Y A364052 Cf. A001222, A062813, A165712, A364049.
%K A364052 nonn,base,hard,more
%O A364052 2,1
%A A364052 _Robert Israel_, Jul 04 2023
%E A364052 a(11) from _Jon E. Schoenfield_, Jul 05 2023
%E A364052 a(12) from _Martin Ehrenstein_, Jul 07 2023
%E A364052 a(13)-a(18) from _Jon E. Schoenfield_, Jul 08 2023
%E A364052 a(19)-a(22) from _Pontus von Brömssen_, Jul 13 2023
%E A364052 a(23)-a(32) from _Bert Dobbelaere_, Jul 20 2023