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A328014 Numbers whose powerful part (A057521) is larger than their powerfree part (A055231). 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A048098 A322109 A122145 * A359869 A034030 A057109
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 01 2019
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)