A359361 Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.

1, 2, 1, 2, 3, 2, 3, 4, 1, 2, 3, 2, 4, 3, 4, 5, 2, 3, 4, 6, 4, 5, 3, 5, 1, 2, 3, 4, 7, 2, 4, 5, 8, 3, 4, 5, 4, 6, 5, 6, 9, 2, 3, 4, 5, 3, 6, 6, 7, 2, 4, 6, 4, 5, 6, 10, 3, 5, 6, 11, 1, 2, 3, 4, 5, 5, 7, 7, 8, 4, 7, 2, 4, 5, 6, 12, 8, 9, 6, 8, 3, 4, 5, 6, 13

Offset: 2

Author

Gus Wiseman, Dec 30 2022

Keywords

• nonn
• tabf

Comments

The partial sums of a sequence (a, b, c, ...) are (a, a+b, a+b+c, ...).

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The partition with Heinz number n is the reversed n-th row of A112798.

Examples

			Triangle begins:
   2: 1
   3: 2
   4: 1 2
   5: 3
   6: 2 3
   7: 4
   8: 1 2 3
   9: 2 4
  10: 3 4
  11: 5
  12: 2 3 4
  13: 6
  14: 4 5
  15: 3 5
  16: 1 2 3 4
For example, the integer partition with Heinz number 90 is (3,2,2,1), so row n = 90 is (3,5,7,8).
		

Crossrefs

Row-lengths are A001222.

The version for standard compositions is A048793, non-reversed A358134.

Last element in each row is A056239.

First element in each row is A061395

Rows are the partial sums of rows of A296150.

Row-sums are A304818.

A reverse version is A358136, row sums A318283, Heinz numbers A358137.

The sorted Heinz numbers of rows are A359397.

A000041 counts partitions, strict A000009.

A112798 lists prime indices, product A003963.

A355536 lists differences of prime indices.

Cf. A000720, A001221, A055396, A261079, A325362.

Programs

Maple

T:= n-> ListTools[PartialSums](sort([seq(numtheory
       [pi](i[1])$i[2], i=ifactors(n)[2])], `>`))[]:
seq(T(n), n=2..50);  # Alois P. Heinz, Jan 01 2023

Mathematica

Table[Accumulate[Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,2,30}]