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A359361
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Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.
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28
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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
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OFFSET
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2,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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).
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MAPLE
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T:= n-> ListTools[PartialSums](sort([seq(numtheory
[pi](i[1])$i[2], i=ifactors(n)[2])], `>`))[]:
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MATHEMATICA
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Table[Accumulate[Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]], {n, 2, 30}]
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CROSSREFS
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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.
The sorted Heinz numbers of rows are A359397.
A355536 lists differences of prime indices.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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