A280171 - OEIS

%I #15 Jan 27 2025 09:59:23

%S 1,2,6,12,40,30,84,120,288,270,990,420,936,1638,840,2160,5100,4410,

%T 8208,5940,3360,6930,12420,10920,14400,19890,28080,27300,48720,43890,

%U 37200,73440,84480,151470,97440,107100,139860,139650,120120,83160,196800,395850,216720,318780,191520,217350,118440,546000,282240,1222650

%N a(n) is the least positive number whose divisors have all possible residues mod n.

%C a(n) <= n!

%C A000203(a(n)) >= n.

%C a(n) is divisible by n.

%H Alois P. Heinz, <a href="/A280171/b280171.txt">Table of n, a(n) for n = 1..100</a>

%e a(4) = 12: the divisors of 12 include 4, 1, 2, 3 == 0, 1, 2, 3 (mod 4).

%e a(5) = 40: the divisors of 40 include 5, 1, 2, 8, 4 == 0, 1, 2, 3, 4 (mod 5).

%p A[1]:= 1:

%p for k from 2 to 10^6 do

%p   V:= numtheory:-divisors(k);

%p   for m from 2 to nops(V) do

%p     if not(assigned(A[m])) and (map(`modp`,V,m) = {$0..m-1}) then

%p       A[m]:= k

%p     fi

%p   od

%p od:

%p seq(A(n),n=1..49);

%t a[n_] := a[n] = If[n == 1, 1, For[k = n, True, k += n, If[Union[Mod[Divisors[k], n]] == Range[0, n-1], Return[k]]]];

%t Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jan 27 2025 *)

%o (PARI) a(n) = my(k=1); while (#Set(apply(x->Mod(x, n), divisors(k))) != n, k++); k; \\ _Michel Marcus_, Jan 27 2025

%Y Cf. A000203.

%K nonn

%O 1,2

%A _Robert Israel_, Dec 27 2016