OFFSET
1,2
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) 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
Prime indices of 180 are (3,2,2,1,1), with section-sum partition (6,3), so a(180) = 65.
The terms together with their prime indices begin:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
7: {4}
11: {5}
10: {1,3}
13: {6}
11: {5}
11: {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[Times@@Prime/@egs[prix[n]], {n, 100}]
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Gus Wiseman, Feb 26 2025
STATUS
approved