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q-powerful numbers. Numbers whose factorization into factors prime(i)/i has no factor of multiplicity 1.
5

%I #8 May 22 2019 22:08:31

%S 1,4,8,9,16,18,25,27,32,36,49,50,54,64,72,75,81,98,100,108,121,125,

%T 128,144,150,162,169,196,200,216,225,242,243,250,256,288,289,300,324,

%U 338,343,361,363,375,392,400,432,441,450,484,486,500,507,512,529,576

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

%C First differs from A070003 in having 1 and lacking 147.

%C Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:

%C 11 = q(1) q(2) q(3) q(5)

%C 50 = q(1)^3 q(2)^2 q(3)^2

%C 360 = q(1)^6 q(2)^3 q(3)

%C Also Matula-Goebel numbers of rooted trees with no terminal subtree appearing at only one place in the tree.

%H Charlie Neder, <a href="/A325661/b325661.txt">Table of n, a(n) for n = 1..1071</a> (Terms <= 100000)

%e The sequence of terms together with their q-signatures begins:

%e 1: {}

%e 4: {2}

%e 8: {3}

%e 9: {2,2}

%e 16: {4}

%e 18: {3,2}

%e 25: {2,2,2}

%e 27: {3,3}

%e 32: {5}

%e 36: {4,2}

%e 49: {4,2}

%e 50: {3,2,2}

%e 54: {4,3}

%e 64: {6}

%e 72: {5,2}

%e 75: {3,3,2}

%e 81: {4,4}

%e 98: {5,2}

%e 100: {4,2,2}

%t difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];

%t Select[Range[100],Count[Length/@Split[difac[#]],1]==0&]

%Y Cf. A001222, A001221, A001694, A056239, A112798, A124010.

%Y Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713.

%Y q-factorization: A324922, A324923, A324924, A325615, A325660.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 13 2019