OFFSET
1,2
COMMENTS
Row-lengths are A051903.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The section-sum partition of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
EXAMPLE
The prime indices of 24 are (2,1,1,1), with sections ((2,1),(1),(1)), so row 24 is (3,1,1).
Triangle begins:
1: (empty)
2: 1
3: 2
4: 1 1
5: 3
6: 3
7: 4
8: 1 1 1
9: 2 2
10: 4
11: 5
12: 3 1
13: 6
14: 5
15: 5
16: 1 1 1 1
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
egs[y_]:=If[y=={}, {}, Table[Total[Select[Union[y], Count[y, #]>=i&]], {i, Max@@Length/@Split[y]}]];
Table[egs[prix[n]], {n, 100}]
CROSSREFS
KEYWORD
nonn,tabf,new
AUTHOR
Gus Wiseman, Feb 28 2025
STATUS
approved