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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 19 12:57 EDT 2019. Contains 327198 sequences. (Running on oeis4.)