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

1,2

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]

Links

Table of n, a(n) for n=1..83.

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

Example

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;

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

Crossrefs

Row sums give A047968.

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

Sequence in context: A052552 A147000 A147486 * A173238 A173284 A278136

Adjacent sequences: A168015 A168016 A168017 * A168019 A168020 A168021

Keyword

nonn,tabf

Author

Omar E. Pol, Nov 22 2009

Extensions

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

Status

approved