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A359361
Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.
28
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
2,2
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.
EXAMPLE
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).
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}]
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.
Sequence in context: A291985 A317192 A053735 * A033667 A033923 A233420
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Dec 30 2022
STATUS
approved