cp's OEIS Frontend

A325609 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.

%I A325609 #10 May 13 2019 01:09:47
%S A325609 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 A325609 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 A325609 11,4,3,1,1,1,36,13,4,3,1,1,1,39,13,4,3,1,1,1,1
%N A325609 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 A325609 Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
%C A325609    11 = q(1) q(2) q(3) q(5)
%C A325609    50 = q(1)^3 q(2)^2 q(3)^2
%C A325609   360 = q(1)^6 q(2)^3 q(3)
%C A325609 Row n is the sequence of nonzero exponents in the q-factorization of n!.
%C A325609 Also the number of terminal subtrees with Matula-Goebel number k of the rooted tree with Matula-Goebel number n!.
%e A325609 We have 10! = q(1)^16 q(2)^6 q(3)^2 q(4), so row n = 10 is (16,6,2,1).
%e A325609 Triangle begins:
%e A325609   {}
%e A325609    1
%e A325609    2  1
%e A325609    4  1
%e A325609    5  2  1
%e A325609    7  3  1
%e A325609    9  3  1  1
%e A325609   12  3  1  1
%e A325609   14  5  1  1
%e A325609   16  6  2  1
%e A325609   17  7  3  1  1
%e A325609   20  8  3  1  1
%e A325609   22  9  3  1  1  1
%e A325609   25  9  3  2  1  1
%e A325609   27 11  4  2  1  1
%e A325609   31 11  4  2  1  1
%e A325609   33 11  4  3  1  1  1
%e A325609   36 13  4  3  1  1  1
%e A325609   39 13  4  3  1  1  1  1
%e A325609   42 14  5  3  1  1  1  1
%t A325609 difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
%t A325609 Table[Length/@Split[difac[n!]],{n,20}]
%Y A325609 Row lengths are A000720.
%Y A325609 Row sums are A325544(n) - 1.
%Y A325609 Column k = 1 is A325543.
%Y A325609 Cf. A056239, A067255, A112798, A118914, A124010.
%Y A325609 Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324935.
%Y A325609 Factorial numbers: A000142, A011371, A022559, A071626, A115627, A325276.
%Y A325609 q-factorization: A324922, A324923, A324924, A325614, A325615, A325660.
%K A325609 nonn,tabf
%O A325609 1,2
%A A325609 _Gus Wiseman_, May 12 2019