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%I #6 Mar 21 2019 17:21:45
%S 1,1,2,1,1,1,2,3,1,1,2,1,1,4,1,1,1,1,1,2,2,1,1,2,3,1,2,3,5,1,1,1,2,1,
%T 1,2,6,1,1,1,4,1,1,2,2,3,1,1,1,1,1,1,4,7,1,1,1,2,2,1,1,1,8,1,1,1,2,3,
%U 1,1,1,2,4,1,1,2,3,5,1,1,2,2,9
%N Irregular triangle read by rows giving the factorization of n into factors q(i) = prime(i)/i, i > 0.
%C Row n is the multiset of Matula-Goebel numbers of all proper terminal subtrees of the rooted tree with Matula-Goebel number n. For example, the rooted tree with Matula-Goebel number 1362 is (o(o)((oo)(oo))), with proper terminal subtrees {o,o,o,o,o,o,(o),(oo),(oo),((oo)(oo))}, which have Matula-Goebel numbers {1,1,1,1,1,1,2,4,4,49}, which is row 1362, as required.
%e Triangle begins:
%e {}
%e 1
%e 1 2
%e 1 1
%e 1 2 3
%e 1 1 2
%e 1 1 4
%e 1 1 1
%e 1 1 2 2
%e 1 1 2 3
%e 1 2 3 5
%e 1 1 1 2
%e 1 1 2 6
%e 1 1 1 4
%e 1 1 2 2 3
%e 1 1 1 1
%e 1 1 4 7
%e 1 1 1 2 2
%e 1 1 1 8
%e 1 1 1 2 3
%e 1 1 1 2 4
%e 1 1 2 3 5
%e 1 1 2 2 9
%e For example, row 65 is {1,1,1,2,2,3,6} because 65 = q(1)^3 q(2)^2 q(3) q(6).
%t difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
%t Table[difac[n],{n,30}]
%Y Cf. A000081, A003963, A061775, A109082, A109129, A112798, A120383, A196050, A317713, A324850, A324922, A324923, A324925, A324931, A324934.
%K nonn,tabf
%O 1,3
%A _Gus Wiseman_, Mar 20 2019
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