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A328014 Numbers whose powerful part (A057521) is larger than their powerfree part (A055231).

%I A328014 #11 Jul 03 2020 14:55:05
%S A328014 4,8,9,12,16,18,24,25,27,32,36,40,45,48,49,50,54,56,63,64,72,75,80,81,
%T A328014 96,98,100,108,112,121,125,128,135,144,147,150,160,162,169,175,176,
%U A328014 180,189,192,196,200,208,216,224,225,240,242,243,245,250,252,256,270
%N A328014 Numbers whose powerful part (A057521) is larger than their powerfree part (A055231).
%C A328014 Differs from A122145(n) at n >= 25.
%C A328014 Cloutier et al. showed that the number of terms of this sequence below x is D0 * x^(3/4) + O(x^(2/3)*log(x)), where D0 is a constant given in A328015.
%H A328014 Amiram Eldar, <a href="/A328014/b328014.txt">Table of n, a(n) for n = 1..10000</a>
%H A328014 Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Cloutier/cloutier2.html">On the powerful and squarefree parts of an integer</a>, Journal of Integer Sequences,  Vol. 17 (2014), Article 14.6.6.
%e A328014 12 is in the sequence since A057521(12) = 4 > A055231(12) = 3.
%t A328014 funp[p_, e_] := If[e > 1, p^e, 1]; pow[n_] := Times @@ (funp @@@ FactorInteger[n]); aQ[n_] := pow[n] > n/pow[n]; Select[Range[1000], aQ]
%o A328014 (PARI) pful(f) = prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); \\ A057521
%o A328014 pfree(f) = for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); \\ A055231
%o A328014 isok(n) = my(f=factor(n)); pful(f) > pfree(f); \\ _Michel Marcus_, Oct 02 2019
%Y A328014 Cf. A057521, A055231, A328015.
%K A328014 nonn
%O A328014 1,1
%A A328014 _Amiram Eldar_, Oct 01 2019