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

1,2

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Henry Bottomley, Some Smarandache-type multiplicative sequences.

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

Example

a(12) = 36/4 = 9.

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 *)

Prog

(Haskell)
a055491 = (^ 2) . a007913  -- Reinhard Zumkeller, Jul 23 2014
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*(f[i, 2]%2))); } \\ Amiram Eldar, Oct 27 2022

Crossrefs

Cf. A056551, A056552.

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

Cf. A013661, A013664.

Sequence in context: A236104 A152205 A129861 * A032523 A350623 A032760

Adjacent sequences: A055488 A055489 A055490 * A055492 A055493 A055494

Keyword

easy,nonn,mult

Author

Henry Bottomley, Jun 28 2000

Status

approved