A325613 - OEIS

%I #6 May 13 2019 01:10:15

%S 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 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 1,2,2,0,0,0,0,0,0,1,4,1,2,2,2,3,1,0,0

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

%e Triangle begins:

%e   {}

%e   1

%e   1 1

%e   2

%e   1 1 1

%e   2 1

%e   2 0 0 1

%e   3

%e   2 2

%e   2 1 1

%e   1 1 1 0 1

%e   3 1

%e   2 1 0 0 0 1

%e   3 0 0 1

%e   2 2 1

%e   4

%e   2 0 0 1 0 0 1

%e   3 2

%e   3 0 0 0 0 0 0 1

%e   3 1 1

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

%t qsig[n_]:=If[n==1,{},With[{ms=difac[n]},Table[Count[ms,i],{i,Max@@ms}]]];

%t Table[qsig[n],{n,30}]

%Y Row lengths are A061395.

%Y Row sums are A196050.

%Y Row-maxima are A109129.

%Y The number whose full prime signature is the n-th row is A324922(n).

%Y Cf. A067255.

%Y Matula-Goebel numbers: A007097, A061775, A109082, A317713.

%Y q-factorization: A324923, A324924, A325613, A325614, A325615, A325660.

%K nonn,tabf

%O 1,4

%A _Gus Wiseman_, May 12 2019