A325615 - OEIS

%I #4 May 13 2019 01:10:28

%S 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,

%T 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,

%U 1,2,3,1,1,1,1,1,5,1,1,2,2,1,1,3,1,1,1

%N Sorted q-signature of n.

%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 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).

%e Triangle begins:

%e   {}

%e   1

%e   1 1

%e   2

%e   1 1 1

%e   1 2

%e   1 2

%e   3

%e   2 2

%e   1 1 2

%e   1 1 1 1

%e   1 3

%e   1 1 2

%e   1 3

%e   1 2 2

%e   4

%e   1 1 2

%e   2 3

%e   1 3

%e   1 1 3

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

%t Table[Sort[Length/@Split[difac[n]]],{n,30}]

%Y Row lengths are A324923.

%Y Row sums are A196050.

%Y Row-maxima are A109129.

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

%K nonn,tabf

%O 1,4

%A _Gus Wiseman_, May 12 2019