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Regular triangle where T(n,k) is the number of factorizations of n into factors > 1 with sum k.
7

%I #17 Oct 22 2018 17:41:58

%S 0,0,1,0,0,1,0,0,0,2,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,2,

%T 0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,

%U 0,0,0,0,2,1,0,0,0,1,0,0,0,0,0,0,0,0,0

%N Regular triangle where T(n,k) is the number of factorizations of n into factors > 1 with sum k.

%e Triangle begins:

%e 0

%e 0 1

%e 0 0 1

%e 0 0 0 2

%e 0 0 0 0 1

%e 0 0 0 0 1 1

%e 0 0 0 0 0 0 1

%e 0 0 0 0 0 2 0 1

%e 0 0 0 0 0 1 0 0 1

%e 0 0 0 0 0 0 1 0 0 1

%e 0 0 0 0 0 0 0 0 0 0 1

%e 0 0 0 0 0 0 2 1 0 0 0 1

%e 0 0 0 0 0 0 0 0 0 0 0 0 1

%e 0 0 0 0 0 0 0 0 1 0 0 0 0 1

%e 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

%e 0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 1

%e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

%e 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1

%e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

%e 0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0 0 1

%e Row 12 {0,0,0,0,0,0,2,1,0,0,0,1} corresponds to the factorizations:

%e . . . . . . (3*4) (2*6) . . . (12)

%e (2*2*3)

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],Total[#]==k&]],{n,20},{k,n}]

%Y Row sums are A001055. Column sums are A002865.

%Y Cf. A069016, A096276, A301987, A319000, A319005, A319057, A319916.

%K nonn,tabl

%O 1,10

%A _Gus Wiseman_, Oct 22 2018