OFFSET
0,5
COMMENTS
A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums.
EXAMPLE
Triangle begins:
1
0 1
0 2 1
0 3 0 1
0 5 3 0 1
0 7 0 0 0 1
0 11 6 4 0 0 1
0 15 0 0 0 0 0 1
0 22 14 0 5 0 0 0 1
0 30 0 10 0 0 0 0 0 1
0 42 25 0 0 6 0 0 0 0 1
0 56 0 0 0 0 0 0 0 0 0 1
0 77 53 30 15 0 7 0 0 0 0 0 1
Row n = 6 counts the following partitions:
. (6) (33) (222) . . (111111)
(51) (321) (2211)
(42) (3111) (21111)
(411) (2211) (111111)
(33) (21111)
(321) (111111)
(3111)
(222)
(2211)
(21111)
(111111)
MATHEMATICA
hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]], {n, 0, 10}, {k, 0, n}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 20 2024
STATUS
approved