A328014 - OEIS

%I #11 Jul 03 2020 14:55:05

%S 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 96,98,100,108,112,121,125,128,135,144,147,150,160,162,169,175,176,

%U 180,189,192,196,200,208,216,224,225,240,242,243,245,250,252,256,270

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

%C Differs from A122145(n) at n >= 25.

%C 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 Amiram Eldar, <a href="/A328014/b328014.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 12 is in the sequence since A057521(12) = 4 > A055231(12) = 3.

%t 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 (PARI) pful(f) = prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); \\ A057521

%o pfree(f) = for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); \\ A055231

%o isok(n) = my(f=factor(n)); pful(f) > pfree(f); \\ _Michel Marcus_, Oct 02 2019

%Y Cf. A057521, A055231, A328015.

%K nonn

%O 1,1

%A _Amiram Eldar_, Oct 01 2019