A325615 Sorted q-signature of n.

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 1, 2, 2, 4, 1, 1, 2, 2, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 2, 1, 4, 2, 2, 2, 1, 1, 3, 3, 3, 1, 4, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1

Offset: 1

Author

Gus Wiseman, May 12 2019

Keywords

• nonn
• tabf

Comments

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)

Row n is the multiset of nonzero multiplicities in the q-factorization of n. For example, row 11 is (1,1,1,1) and row 360 is (1,3,6).

Examples

			Triangle begins:
  {}
  1
  1 1
  2
  1 1 1
  1 2
  1 2
  3
  2 2
  1 1 2
  1 1 1 1
  1 3
  1 1 2
  1 3
  1 2 2
  4
  1 1 2
  2 3
  1 3
  1 1 3
		

Crossrefs

Row lengths are A324923.

Row sums are A196050.

Row-maxima are A109129.

Cf. A118914, A324922, A324924, A324931, A324934, A325608, A325609, A325613, A325614, A325660.

Programs

Mathematica

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Sort[Length/@Split[difac[n]]],{n,30}]