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A326037
Heinz numbers of uniform perfect integer partitions.
3
1, 2, 4, 6, 8, 16, 32, 42, 64, 100, 128, 256, 512, 798, 1024, 2048, 2744, 4096, 8192, 16384, 32768, 42294, 52900, 65536
OFFSET
1,2
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition of n is uniform if all parts appear with the same multiplicity, and perfect if every nonnegative integer up to n is the sum of a unique submultiset.
The enumeration of these partitions by sum is given by A089723.
FORMULA
Intersection of A072774 (uniform), A299702 (knapsack), and A325781 (complete).
EXAMPLE
The sequence of all uniform perfect integer partitions together with their Heinz numbers begins:
1: ()
2: (1)
4: (11)
6: (21)
8: (111)
16: (1111)
32: (11111)
42: (421)
64: (111111)
100: (3311)
128: (1111111)
256: (11111111)
512: (111111111)
798: (8421)
1024: (1111111111)
2048: (11111111111)
2744: (444111)
4096: (111111111111)
8192: (1111111111111)
16384: (11111111111111)
MATHEMATICA
hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
Select[Range[1000], SameQ@@Last/@FactorInteger[#]&&Sort[hwt/@Divisors[#]]==Range[0, hwt[#]]&]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 04 2019
STATUS
approved