A280171 a(n) is the least positive number whose divisors have all possible residues mod n.

1, 2, 6, 12, 40, 30, 84, 120, 288, 270, 990, 420, 936, 1638, 840, 2160, 5100, 4410, 8208, 5940, 3360, 6930, 12420, 10920, 14400, 19890, 28080, 27300, 48720, 43890, 37200, 73440, 84480, 151470, 97440, 107100, 139860, 139650, 120120, 83160, 196800, 395850, 216720, 318780, 191520, 217350, 118440, 546000, 282240, 1222650

Offset

1,2

Comments

a(n) <= n!

A000203(a(n)) >= n.

a(n) is divisible by n.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..100

Example

a(4) = 12: the divisors of 12 include 4, 1, 2, 3 == 0, 1, 2, 3 (mod 4).
a(5) = 40: the divisors of 40 include 5, 1, 2, 8, 4 == 0, 1, 2, 3, 4 (mod 5).

Maple

A[1]:= 1:
for k from 2 to 10^6 do
  V:= numtheory:-divisors(k);
  for m from 2 to nops(V) do
    if not(assigned(A[m])) and (map(`modp`, V, m) = {$0..m-1}) then
      A[m]:= k
    fi
  od
od:
seq(A(n), n=1..49);

Mathematica

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]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 27 2025 *)

Prog

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

Crossrefs

Cf. A000203.

Sequence in context: A162589 A225957 A123045 * A327879 A094261 A080497

Adjacent sequences: A280168 A280169 A280170 * A280172 A280173 A280174

Keyword

nonn

Author

Robert Israel, Dec 27 2016

Status

approved