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.
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[#]]&]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 04 2019
STATUS
approved