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

1,2

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.

Links

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

Example

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}

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&]

Crossrefs

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

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

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

Sequence in context: A339744 A226385 A070003 * A073539 A090779 A166402

Adjacent sequences: A325658 A325659 A325660 * A325662 A325663 A325664

Keyword

nonn

Author

Gus Wiseman, May 13 2019

Status

approved