A197863 - OEIS

%I #32 Sep 01 2023 02:55:28

%S 1,4,9,4,25,36,49,8,9,100,121,36,169,196,225,16,289,36,361,100,441,

%T 484,529,72,25,676,27,196,841,900,961,32,1089,1156,1225,36,1369,1444,

%U 1521,200,1681,1764,1849,484,225,2116,2209,144,49,100,2601,676,2809,108,3025

%N Smallest powerful number that is a multiple of n.

%H Reinhard Zumkeller, <a href="/A197863/b197863.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F 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

%F a(n) = n * A055231(n). - _Amiram Eldar_, Sep 01 2023

%t 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 *)

%t f[p_, e_] := p^Max[e, 2]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jul 09 2022 *)

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

%o (Haskell)

%o a197863 n = product $

%o    zipWith (^) (a027748_row n) (map (max 2) $ a124010_row n)

%o -- _Reinhard Zumkeller_, Jan 06 2012

%Y Cf. A055231, A087320, A001694, A086463.

%Y Cf. A027748, A124010, A007948.

%K nonn,easy,mult

%O 1,2

%A _Franklin T. Adams-Watters_, Oct 18 2011