A035310 -id:A035310 - cp's OEIS frontend

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

Alford Arnold, May 23 2004

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

			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;
...

Cf. A001055, A050322.

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

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

More terms from Alford Arnold, Nov 19 2005

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

A129305 Encodes multisets of least prime signatures in reverse-lex order: replace A036035 with A080688 then calculate all possible factorizations of the resulting values, recode each factor using A064553(n) and then multiply the terms.

Original entry on oeis.org

1, 2, 4, 5, 6, 11, 8, 10, 17, 12, 15, 22, 31, 42, 69, 77, 86, 109, 16, 20, 25, 34, 47, 24, 30, 44, 55, 51, 62, 83, 36, 45, 66, 76, 95, 121, 93, 118, 149, 84, 105, 138, 154, 172, 215, 253, 201, 217, 218, 277, 546, 834, 861, 897, 994, 1001, 1118, 1529, 1633, 1763, 1041

Offset: 0

Alford Arnold, May 02 2007

nonn

Sequence A035310 counts the values in each subtable and illustrates relationships with A000041, A000079, A000110 etc. Sequence A096443 counts the values associated with each least prime signature. (Cf. A025487 and A036035.)

			The encoded values can be arranged in tabular form based on the number of factors and the associated numeric partitions as indicated below:
2..................................................
.....4.....5........................................
.....6.....11........................................
...............8.....10.....17.........................
...............12....15.....31.........................
.....................22..............................
...............42....69.....109.........................
.....................77..............................
.....................86..............................
.................................16.....20.....25.....34.....47
.................................24.....30.....55.....51.....83
........................................44............62.....
.................................36.....45.....95.....93.....149
........................................66.....121...118.....
........................................76...............
.................................84.....105.....215.....201.....277
........................................138.....253.....217.....
........................................154.............218.....
........................................172...............
................................546.....834.....1529.....1041.....1289
........................................861.....1633.....1138.....
........................................897.....1763.....1253.....
........................................994..............1417.....
........................................1001...............
........................................1118...............

Cf. A025487, A035310, A036035, A064553, A080688, A096443.

cp's OEIS Frontend

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.

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