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

1, 4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 75, 81, 98, 100, 108, 121, 125, 128, 144, 150, 162, 169, 196, 200, 216, 225, 242, 243, 250, 256, 288, 289, 300, 324, 338, 343, 361, 363, 375, 392, 400, 432, 441, 450, 484, 486, 500, 507, 512, 529, 576

Offset: 1

Author

Gus Wiseman, May 13 2019

Keywords

• nonn

Comments

First differs from A070003 in having 1 and lacking 147.

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

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

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

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

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

Examples

			The sequence of terms together with their q-signatures begins:
    1: {}
    4: {2}
    8: {3}
    9: {2,2}
   16: {4}
   18: {3,2}
   25: {2,2,2}
   27: {3,3}
   32: {5}
   36: {4,2}
   49: {4,2}
   50: {3,2,2}
   54: {4,3}
   64: {6}
   72: {5,2}
   75: {3,3,2}
   81: {4,4}
   98: {5,2}
  100: {4,2,2}
		

Links

• Charlie Neder, Table of n, a(n) for n = 1..1071 (Terms <= 100000)

Crossrefs

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

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

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

Programs

Mathematica

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Select[Range[100],Count[Length/@Split[difac[#]],1]==0&]