A325609 - OEIS

%I #10 May 13 2019 01:09:47

%S 1,2,1,4,1,5,2,1,7,3,1,9,3,1,1,12,3,1,1,14,5,1,1,16,6,2,1,17,7,3,1,1,

%T 20,8,3,1,1,22,9,3,1,1,1,25,9,3,2,1,1,27,11,4,2,1,1,31,11,4,2,1,1,33,

%U 11,4,3,1,1,1,36,13,4,3,1,1,1,39,13,4,3,1,1,1,1

%N Unsorted q-signature of n!. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the factorization of n! into factors q(i) = prime(i)/i.

%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 sequence of nonzero exponents in the q-factorization of n!.

%C Also the number of terminal subtrees with Matula-Goebel number k of the rooted tree with Matula-Goebel number n!.

%e We have 10! = q(1)^16 q(2)^6 q(3)^2 q(4), so row n = 10 is (16,6,2,1).

%e Triangle begins:

%e   {}

%e    1

%e    2  1

%e    4  1

%e    5  2  1

%e    7  3  1

%e    9  3  1  1

%e   12  3  1  1

%e   14  5  1  1

%e   16  6  2  1

%e   17  7  3  1  1

%e   20  8  3  1  1

%e   22  9  3  1  1  1

%e   25  9  3  2  1  1

%e   27 11  4  2  1  1

%e   31 11  4  2  1  1

%e   33 11  4  3  1  1  1

%e   36 13  4  3  1  1  1

%e   39 13  4  3  1  1  1  1

%e   42 14  5  3  1  1  1  1

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

%t Table[Length/@Split[difac[n!]],{n,20}]

%Y Row lengths are A000720.

%Y Row sums are A325544(n) - 1.

%Y Column k = 1 is A325543.

%Y Cf. A056239, A067255, A112798, A118914, A124010.

%Y Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324935.

%Y Factorial numbers: A000142, A011371, A022559, A071626, A115627, A325276.

%Y q-factorization: A324922, A324923, A324924, A325614, A325615, A325660.

%K nonn,tabf

%O 1,2

%A _Gus Wiseman_, May 12 2019