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A197863
Smallest powerful number that is a multiple of n.
13
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
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
KEYWORD
nonn,easy,mult
AUTHOR
STATUS
approved