A055491 Smallest square divisible by n divided by largest square which divides n.

1, 4, 9, 1, 25, 36, 49, 4, 1, 100, 121, 9, 169, 196, 225, 1, 289, 4, 361, 25, 441, 484, 529, 36, 1, 676, 9, 49, 841, 900, 961, 4, 1089, 1156, 1225, 1, 1369, 1444, 1521, 100, 1681, 1764, 1849, 121, 25, 2116, 2209, 9, 1, 4, 2601, 169, 2809, 36, 3025, 196, 3249, 3364

Offset: 1

Author

Henry Bottomley, Jun 28 2000

Keywords

• easy
• nonn
• mult

Examples

			a(12) = 36/4 = 9.
		

Links

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

Crossrefs

Cf. A056551, A056552.

Cf. A000188, A000290, A007913, A008833, A019554, A053143.

Cf. A013661, A013664.

Programs

Haskell

a055491 = (^ 2) . a007913  -- Reinhard Zumkeller, Jul 23 2014

Mathematica

With[{sqs=Range[100]^2},Table[SelectFirst[sqs,Divisible[#,n]&]/ SelectFirst[ Reverse[sqs],Divisible[n,#]&],{n,60}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 18 2018 *)
f[p_, e_] := p^(2 * Mod[e, 2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2*(f[i,2]%2)));} \\ Amiram Eldar, Oct 27 2022

Formula

If n is written as Product(Pj^Ej) then a(n) = Product(Pj^(2*(Ej mod 2))).

a(n) = A053143(n)/A008833(n) = A007913(n)^2 = (A019554(n)/A000188(n))^2 = A000290(n)/A008833(n)^2.

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(6)/(3*zeta(2))) = 2*Pi^4/945 = 0.206156... . - Amiram Eldar, Oct 27 2022

Dirichlet g.f.: zeta(s-2) * zeta(2*s) / zeta(2*s-4). - Amiram Eldar, Sep 16 2023