A168018 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n.

1, 2, 1, 3, 1, 5, 2, 1, 7, 1, 11, 3, 2, 1, 15, 1, 22, 5, 2, 1, 30, 3, 1, 42, 7, 2, 1, 56, 1, 77, 11, 5, 3, 2, 1, 101, 1, 135, 15, 2, 1, 176, 7, 3, 1, 231, 22, 5, 2, 1, 297, 1, 385, 30, 11, 3, 2, 1, 490, 1, 627, 42, 7, 5, 2, 1, 792, 15, 3, 1, 1002, 56, 2, 1, 1255, 1, 1575, 77, 22, 11, 5, 3, 2

Offset: 1

Author

Omar E. Pol, Nov 22 2009

Keywords

• nonn
• tabf

Comments

Positive values of triangle A168021.

Note that column 1 lists the numbers of partitions A000041(n).

Row n has A000005(n) terms.

Also, it appears that row n lists the partition numbers of the divisors of n, in decreasing order. [Omar E. Pol, Nov 23 2009]

Examples

			For example:
Consider row 8: (22, 5, 2, 1). The divisors of 8 are 1, 2, 4, 8 (see A027750). Also, there are 22 partitions of 8 into parts divisible by 1 (A000041(8)=22); 5 partitions of 8 into parts divisible by 2: {(8),(6+2),(4+4),(4+2+2),(2+2+2+2)}; 2 partitions of 8 into parts divisible by 4: {(8),(4+4)}; and 1 partition of 8 into parts divisible by 8. Then row 8 is formed by 22, 5, 2, 1.
Triangle begins:
1;
2, 1;
3, 1;
5, 2, 1;
7, 1;
11, 3, 2, 1;
15, 1;
22, 5, 2, 1;
30, 3, 1;
42, 7, 2, 1;
56, 1;
77, 11, 5, 3, 2, 1;
		

Links

• Omar E. Pol, Illustration of the partitions of n, for n = 1 .. 9

Crossrefs

Row sums give A047968.

Cf. A000005, A000041, A027750, A135010, A138121, A168016, A168017, A168019, A168020, A168021.

Programs

Maple

A168018 := proc(n) local dvs,p,i,d,a,pp,divs,par; dvs := sort(convert(numtheory[divisors](n),list)) ; p := combinat[partition](n) ; for i from 1 to nops(dvs) do d := op(i,dvs) ; a := 0 ; for pp in p do divs := true; for par in pp do if par mod d <> 0 then divs := false; end if; end do ; if divs then a := a+1 ; end if; end do ; printf("%d,",a) ; end do ; end proc: for n from 1 to 40 do A168018(n) ; end do : # R. J. Mathar, Feb 05 2010

Extensions

Terms beyond row 12 from R. J. Mathar, Feb 05 2010