cp's OEIS Frontend

A325613 Full q-signature of n. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the q-factorization of n.

%I A325613 #6 May 13 2019 01:10:15
%S A325613 1,1,1,2,1,1,1,2,1,2,0,0,1,3,2,2,2,1,1,1,1,1,0,1,3,1,2,1,0,0,0,1,3,0,
%T A325613 0,1,2,2,1,4,2,0,0,1,0,0,1,3,2,3,0,0,0,0,0,0,1,3,1,1,3,1,0,1,2,1,1,0,
%U A325613 1,2,2,0,0,0,0,0,0,1,4,1,2,2,2,3,1,0,0
%N A325613 Full q-signature of n. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the q-factorization of n.
%C A325613 Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
%C A325613    11 = q(1) q(2) q(3) q(5)
%C A325613    50 = q(1)^3 q(2)^2 q(3)^2
%C A325613   360 = q(1)^6 q(2)^3 q(3)
%C A325613 Also the number of terminal subtrees with Matula-Goebel number k of the rooted tree with Matula-Goebel number n.
%e A325613 Triangle begins:
%e A325613   {}
%e A325613   1
%e A325613   1 1
%e A325613   2
%e A325613   1 1 1
%e A325613   2 1
%e A325613   2 0 0 1
%e A325613   3
%e A325613   2 2
%e A325613   2 1 1
%e A325613   1 1 1 0 1
%e A325613   3 1
%e A325613   2 1 0 0 0 1
%e A325613   3 0 0 1
%e A325613   2 2 1
%e A325613   4
%e A325613   2 0 0 1 0 0 1
%e A325613   3 2
%e A325613   3 0 0 0 0 0 0 1
%e A325613   3 1 1
%t A325613 difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
%t A325613 qsig[n_]:=If[n==1,{},With[{ms=difac[n]},Table[Count[ms,i],{i,Max@@ms}]]];
%t A325613 Table[qsig[n],{n,30}]
%Y A325613 Row lengths are A061395.
%Y A325613 Row sums are A196050.
%Y A325613 Row-maxima are A109129.
%Y A325613 The number whose full prime signature is the n-th row is A324922(n).
%Y A325613 Cf. A067255.
%Y A325613 Matula-Goebel numbers: A007097, A061775, A109082, A317713.
%Y A325613 q-factorization: A324923, A324924, A325613, A325614, A325615, A325660.
%K A325613 nonn,tabf
%O A325613 1,4
%A A325613 _Gus Wiseman_, May 12 2019