A328014 - OEIS

A328014

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

4, 8, 9, 12, 16, 18, 24, 25, 27, 32, 36, 40, 45, 48, 49, 50, 54, 56, 63, 64, 72, 75, 80, 81, 96, 98, 100, 108, 112, 121, 125, 128, 135, 144, 147, 150, 160, 162, 169, 175, 176, 180, 189, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 245, 250, 252, 256, 270

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OFFSET

1,1

COMMENTS

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

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.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.

EXAMPLE

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

MATHEMATICA

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]

PROG

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

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

isok(n) = my(f=factor(n)); pful(f) > pfree(f); \\ Michel Marcus, Oct 02 2019

CROSSREFS

Cf. A057521, A055231, A328015.

Sequence in context: A048098 A322109 A122145 * A359869 A034030 A057109

Adjacent sequences: A328011 A328012 A328013 * A328015 A328016 A328017

KEYWORD

nonn

AUTHOR

Amiram Eldar, Oct 01 2019

STATUS

approved