1,2

Sequence A050322 calculates factorizations indexed by prime signatures: A001055(A025487) For example, A050322(36)=A01055(A025487(36))= 74 and A050322(43) = A001055(A024487(43)) = 92

Note that A093936 can be readily extended by combining appropriate values from A096443. Row sums of A093936 yield A035310 and embeded sequences include A000041, A035098 and A000110. - Alford Arnold (Alford1940(AT)aol.com), Nov 19 2005

a(19) = 166 because A001055(840) + A001055(1260) = 74 + 92

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). T(4,1)=A001055(16)=5. T(4,2)=A001055(24)+A001055(36)

=7+9=16. T(4,3)=A001055(60)=11. T(4,4)=A001055(210)=15.

Table starts

1;

2, 2;

3, 4, 5;

5, 16, 11, 15;

7, 28, 47, 36, 52;

11, 79, 156, 166, 135, 203;

15, 134, 408, 588, 667, 566, 877;

22, 328, 1057, 2358, 2517, 2978, 2610, 4140;

30, 536, 3036, 6181, 10726, 11913, 14548, 13082, 21147;

42, 1197, 6826, 21336, 40130, 53690, 61421, 76908, 70631, 115975;

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 (mathar(AT)strw.leidenuniv.nl), Jul 27 2007

Cf. A001055, A050322.

Cf. A000041, A000110, A035098, A035310, A096443.

Sequence in context: A126442 A129306 A114094 this_sequence A119353 A140859 A072586

Adjacent sequences: A093933 A093934 A093935 this_sequence A093937 A093938 A093939

nonn,tabl

Alford Arnold (Alford1940(AT)aol.com), May 23 2004

More terms from Alford Arnold (Alford1940(AT)aol.com), Nov 19 2005

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 27 2007