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A325661 q-powerful numbers. Numbers whose factorization into factors prime(i)/i has no factor of multiplicity 1.

%I A325661 #8 May 22 2019 22:08:31
%S A325661 1,4,8,9,16,18,25,27,32,36,49,50,54,64,72,75,81,98,100,108,121,125,
%T A325661 128,144,150,162,169,196,200,216,225,242,243,250,256,288,289,300,324,
%U A325661 338,343,361,363,375,392,400,432,441,450,484,486,500,507,512,529,576
%N A325661 q-powerful numbers. Numbers whose factorization into factors prime(i)/i has no factor of multiplicity 1.
%C A325661 First differs from A070003 in having 1 and lacking 147.
%C A325661 Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
%C A325661    11 = q(1) q(2) q(3) q(5)
%C A325661    50 = q(1)^3 q(2)^2 q(3)^2
%C A325661   360 = q(1)^6 q(2)^3 q(3)
%C A325661 Also Matula-Goebel numbers of rooted trees with no terminal subtree appearing at only one place in the tree.
%H A325661 Charlie Neder, <a href="/A325661/b325661.txt">Table of n, a(n) for n = 1..1071</a> (Terms <= 100000)
%e A325661 The sequence of terms together with their q-signatures begins:
%e A325661     1: {}
%e A325661     4: {2}
%e A325661     8: {3}
%e A325661     9: {2,2}
%e A325661    16: {4}
%e A325661    18: {3,2}
%e A325661    25: {2,2,2}
%e A325661    27: {3,3}
%e A325661    32: {5}
%e A325661    36: {4,2}
%e A325661    49: {4,2}
%e A325661    50: {3,2,2}
%e A325661    54: {4,3}
%e A325661    64: {6}
%e A325661    72: {5,2}
%e A325661    75: {3,3,2}
%e A325661    81: {4,4}
%e A325661    98: {5,2}
%e A325661   100: {4,2,2}
%t A325661 difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
%t A325661 Select[Range[100],Count[Length/@Split[difac[#]],1]==0&]
%Y A325661 Cf. A001222, A001221, A001694, A056239, A112798, A124010.
%Y A325661 Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713.
%Y A325661 q-factorization: A324922, A324923, A324924, A325615, A325660.
%K A325661 nonn
%O A325661 1,2
%A A325661 _Gus Wiseman_, May 13 2019