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A371955
Numbers with triquanimous prime indices.
3
8, 27, 36, 48, 64, 125, 150, 180, 200, 216, 240, 288, 320, 343, 384, 441, 490, 512, 567, 588, 630, 700, 729, 756, 784, 810, 840, 900, 972, 1000, 1008, 1080, 1120, 1200, 1296, 1331, 1344, 1440, 1600, 1694, 1728, 1792, 1815, 1920, 2156, 2178, 2197, 2304, 2310
OFFSET
1,1
COMMENTS
A finite multiset of numbers is defined to be triquanimous iff it can be partitioned into three multisets with equal sums.
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.
EXAMPLE
The terms together with their prime indices begin:
8: {1,1,1}
27: {2,2,2}
36: {1,1,2,2}
48: {1,1,1,1,2}
64: {1,1,1,1,1,1}
125: {3,3,3}
150: {1,2,3,3}
180: {1,1,2,2,3}
200: {1,1,1,3,3}
216: {1,1,1,2,2,2}
240: {1,1,1,1,2,3}
288: {1,1,1,1,1,2,2}
320: {1,1,1,1,1,1,3}
343: {4,4,4}
384: {1,1,1,1,1,1,1,2}
441: {2,2,4,4}
490: {1,3,4,4}
512: {1,1,1,1,1,1,1,1,1}
567: {2,2,2,2,4}
588: {1,1,2,4,4}
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]]}]];
Select[Range[1000], Select[facs[#], Length[#]==3&&SameQ@@hwt/@#&]!={}&]
CROSSREFS
These are the Heinz numbers of the partitions counted by A002220.
For biquanimous we have A357976, counted by A002219.
For non-biquanimous we have A371731, counted by A371795, even case A006827.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A371783 counts k-quanimous partitions.
Sequence in context: A373373 A213519 A352423 * A066215 A086213 A175050
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 19 2024
STATUS
approved