%I #15 Jul 09 2017 18:30:09
%S 1,2,2,3,4,5,5,16,11,15,7,28,47,36,52,11,79,156,166,135,203,15,134,
%T 408,588,667,566,877,22,328,1057,2358,2517,2978,2610,4140,30,536,3036,
%U 6181,10726,11913,14548,13082,21147,42,1197,6826,21336,40130,53690,61421
%N Table T(n,k) read by rows which contains in row n and column k the sum of A001055(A036035(n,j)) over all column indices j where A036035(n,j) has k distinct prime factors.
%C Sequence A050322 calculates factorizations indexed by prime signatures: A001055(A025487) For example, A050322(36) = A001055(A025487(36)) = 74 and A050322(43) = A001055(A024487(43)) = 92.
%C Note that A093936 can be readily extended by combining appropriate values from A096443. Row sums of A093936 yield A035310 and embedded sequences include A000041, A035098 and A000110. - _Alford Arnold_, Nov 19 2005
%e a(19) = 166 because A001055(840) + A001055(1260) = 74 + 92.
%e Row n=4 of A036035 contains 16=2^4, 24=2^3*3, 36=2^2*3^2, 60=2^2*3*5 and 210=2*3*5*7. The 16 has k=1 distinct prime factor; 24 and 36 have k=2 distinct prime factors; 60 has k=3 distinct prime factors; 210 has k=4 distinct prime factors (see A001221).
%e T(4,1)=A001055(16)=5.
%e T(4,2)=A001055(24)+A001055(36)=7+9=16.
%e T(4,3)=A001055(60)=11.
%e T(4,4)=A001055(210)=15.
%e Table starts
%e 1;
%e 2, 2;
%e 3, 4, 5;
%e 5, 16, 11, 15;
%e 7, 28, 47, 36, 52;
%e 11, 79, 156, 166, 135, 203;
%e 15, 134, 408, 588, 667, 566, 877;
%e 22, 328, 1057, 2358, 2517, 2978, 2610, 4140;
%e 30, 536, 3036, 6181, 10726, 11913, 14548, 13082, 21147;
%e 42, 1197, 6826, 21336, 40130, 53690, 61421, 76908, 70631, 115975;
%e ...
%p A036035 := proc(n) local pr,L,a ; a := [] ; pr := combinat[partition](n) ; for L in pr do mul(ithprime(i)^op(-i,L),i=1..nops(L)) ; a := [op(a),%] ; od ; RETURN(a) ; end: A001221 := proc(n) local ifacts ; ifacts := ifactors(n)[2] ; nops(ifacts) ; end: listProdRep := proc(n,mincomp) local dvs,resul,f,i,rli ; resul := 0 ; if n = 1 then RETURN(1) elif n >= mincomp then dvs := numtheory[divisors](n) ; for i from 1 to nops(dvs) do f := op(i,dvs) ; if f =n and f >= mincomp then resul := resul+1 ; elif f >= mincomp then rli := listProdRep(n/f,f) ; resul := resul+rli ; fi ; od ; fi ; RETURN(resul) ; end: A001055 := proc(n) listProdRep(n,2) ; end: A093936 := proc(n,k) local a, a036035,j ; a := 0 ; a036035 := A036035(n) ; for j in a036035 do if A001221(j) = k then a := a+A001055(j) ; fi ; od ; RETURN(a) ; end: for n from 1 to 10 do for k from 1 to n do printf("%d,",A093936(n,k)) ; od : od : # _R. J. Mathar_, Jul 27 2007
%Y Cf. A001055, A050322.
%Y Cf. A000041, A000110, A035098, A035310, A096443.
%K nonn,tabl
%O 1,2
%A _Alford Arnold_, May 23 2004
%E More terms from _Alford Arnold_, Nov 19 2005
%E More terms from _R. J. Mathar_, Jul 27 2007