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

Author

Franklin T. Adams-Watters, Oct 18 2011

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A055231, A087320, A001694, A086463.

Cf. A027748, A124010, A007948.

Programs

Haskell

a197863 n = product $
   zipWith (^) (a027748_row n) (map (max 2) $ a124010_row n)
-- Reinhard Zumkeller, Jan 06 2012

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 *)

PARI

a(n)=local(fm=factor(n));prod(k=1,matsize(fm)[1],fm[k,1]^max(fm[k,2],2))

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