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A326037 Heinz numbers of uniform perfect integer partitions.

%I A326037 #4 Jun 04 2019 08:37:40
%S A326037 1,2,4,6,8,16,32,42,64,100,128,256,512,798,1024,2048,2744,4096,8192,
%T A326037 16384,32768,42294,52900,65536
%N A326037 Heinz numbers of uniform perfect integer partitions.
%C A326037 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A326037 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.
%C A326037 The enumeration of these partitions by sum is given by A089723.
%F A326037 Intersection of A072774 (uniform), A299702 (knapsack), and A325781 (complete).
%e A326037 The sequence of all uniform perfect integer partitions together with their Heinz numbers begins:
%e A326037       1: ()
%e A326037       2: (1)
%e A326037       4: (11)
%e A326037       6: (21)
%e A326037       8: (111)
%e A326037      16: (1111)
%e A326037      32: (11111)
%e A326037      42: (421)
%e A326037      64: (111111)
%e A326037     100: (3311)
%e A326037     128: (1111111)
%e A326037     256: (11111111)
%e A326037     512: (111111111)
%e A326037     798: (8421)
%e A326037    1024: (1111111111)
%e A326037    2048: (11111111111)
%e A326037    2744: (444111)
%e A326037    4096: (111111111111)
%e A326037    8192: (1111111111111)
%e A326037   16384: (11111111111111)
%t A326037 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A326037 Select[Range[1000],SameQ@@Last/@FactorInteger[#]&&Sort[hwt/@Divisors[#]]==Range[0,hwt[#]]&]
%Y A326037 Cf. A002033, A047966, A070941, A072774, A108917, A126796, A276024, A299702.
%Y A326037 Cf. A325780, A325781, A326020, A326035, A326036.
%K A326037 nonn,more
%O A326037 1,2
%A A326037 _Gus Wiseman_, Jun 04 2019