A364052 - OEIS

%I #33 Jul 20 2023 12:12:33

%S 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 94,97,104,107,113,118,84

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

%e 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 f:= proc(n) local d,S,V,k;

%p   V:= {};

%p   for d from 1 to n do

%p     S:= select(t -> t[-1] <> 0, combinat:-permute([$0..n-1],d));

%p     S:= map(proc(t) local i;  numtheory:-bigomega(add(t[i]*n^(i-1),i=1..d)) end proc, S);

%p     V:= V union convert(S,set);

%p   od;

%p   min({$1..1+max(V)} minus V)

%p end proc:

%p map(f, [$2..10]);

%Y Cf. A001222, A062813, A165712, A364049.

%K nonn,base,hard,more

%O 2,1

%A _Robert Israel_, Jul 04 2023

%E a(11) from _Jon E. Schoenfield_, Jul 05 2023

%E a(12) from _Martin Ehrenstein_, Jul 07 2023

%E a(13)-a(18) from _Jon E. Schoenfield_, Jul 08 2023

%E a(19)-a(22) from _Pontus von Brömssen_, Jul 13 2023

%E a(23)-a(32) from _Bert Dobbelaere_, Jul 20 2023