A093936 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.

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

Offset

1,2

Comments

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.

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

Links

Table of n, a(n) for n=1..52.

Example

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;
...

Maple

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

Crossrefs

Cf. A001055, A050322.

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

Sequence in context: A324325 A318284 A241321 * A329904 A331307 A331306

Adjacent sequences: A093933 A093934 A093935 * A093937 A093938 A093939

Keyword

nonn,tabl

Author

Alford Arnold, May 23 2004

Extensions

More terms from Alford Arnold, Nov 19 2005

More terms from R. J. Mathar, Jul 27 2007

Status

approved