A053143 Smallest square divisible by n.

1, 4, 9, 4, 25, 36, 49, 16, 9, 100, 121, 36, 169, 196, 225, 16, 289, 36, 361, 100, 441, 484, 529, 144, 25, 676, 81, 196, 841, 900, 961, 64, 1089, 1156, 1225, 36, 1369, 1444, 1521, 400, 1681, 1764, 1849, 484, 225, 2116, 2209, 144, 49, 100, 2601, 676, 2809

Offset: 1

Author

Henry Bottomley, Feb 28 2000

Keywords

• easy
• nonn
• mult

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• Henry Bottomley, Some Smarandache-type multiplicative sequences.

Crossrefs

Cf. A000188, A007913, A008833, A019554.

Programs

Mathematica

Table[i = 1; While[Mod[i^2, n] > 0, i++]; i^2, {n, 100}] (* T. D. Noe, Oct 30 2011 *)
f[p_, e_] := p^(e + Mod[e, 2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

PARI

a(n) = n*core(n); /* Joerg Arndt, Aug 02 2012 */

Formula

a(n) = n*A007913(n) = A019554(n)^2 = n^2/A008833(n) = (n/A000188(n))^2.

Multiplicative with p^e -> p^(e + e mod 2), p prime. - Reinhard Zumkeller, Feb 09 2003

Dirichlet g.f.: zeta(2s-2)*zeta(s-2)/zeta(2s-4). - R. J. Mathar, Oct 31 2011

Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / 45. - Vaclav Kotesovec, Feb 08 2019

Sum_{n>=1} 1/a(n) = 5/2. - Amiram Eldar, Jul 29 2022