A197863 Smallest powerful number that is a multiple of n.

1, 4, 9, 4, 25, 36, 49, 8, 9, 100, 121, 36, 169, 196, 225, 16, 289, 36, 361, 100, 441, 484, 529, 72, 25, 676, 27, 196, 841, 900, 961, 32, 1089, 1156, 1225, 36, 1369, 1444, 1521, 200, 1681, 1764, 1849, 484, 225, 2116, 2209, 144, 49, 100, 2601, 676, 2809, 108, 3025

Offset

1,2

Links

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

Formula

Multiplicative with a(p^e) = p^max(e,2).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (2*p-1)/(p^2*(p-1))) = 2.71098009471568319328... . - Amiram Eldar, Jul 29 2022

Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/18) * Product_{p prime} (1 - 2/p^2 + 2/p^4 - 1/p^5) = 0.2165355664... . - Amiram Eldar, Nov 19 2022

a(n) = n * A055231(n). - Amiram Eldar, Sep 01 2023

Mathematica

With[{pwrnos=Join[{1}, Select[Range[5000], Min[Transpose[ FactorInteger[#]] [[2]]]>1&]]}, Flatten[Table[Select[pwrnos, Divisible[#, n]&, 1], {n, 60}]]] (* Harvey P. Dale, Aug 14 2012 *)
f[p_, e_] := p^Max[e, 2]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)

Prog

(PARI) a(n)=local(fm=factor(n)); prod(k=1, matsize(fm)[1], fm[k, 1]^max(fm[k, 2], 2))
(Haskell)
a197863 n = product $
   zipWith (^) (a027748_row n) (map (max 2) $ a124010_row n)
-- Reinhard Zumkeller, Jan 06 2012

Crossrefs

Cf. A055231, A087320, A001694, A086463.

Cf. A027748, A124010, A007948.

Sequence in context: A235323 A345304 A078615 * A087320 A087321 A053143

Adjacent sequences: A197860 A197861 A197862 * A197864 A197865 A197866

Keyword

nonn,easy,mult

Author

Franklin T. Adams-Watters, Oct 18 2011

Status

approved