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A325764 Heinz numbers of integer partitions whose distinct consecutive subsequences have distinct sums that cover an initial interval of positive integers.

%I A325764 #4 May 21 2019 22:05:20
%S A325764 1,2,4,6,8,16,18,20,32,54,56,64,100,128,162,176,256,392,416,486,500,
%T A325764 512,1024,1088,1458,1936,2048,2432,2500,2744,4096,4374,5408,5888,8192,
%U A325764 12500,13122,14848,16384,18496,19208,21296,31744,32768,39366,46208,62500,65536
%N A325764 Heinz numbers of integer partitions whose distinct consecutive subsequences have distinct sums that cover an initial interval of positive integers.
%C A325764 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325764 The enumeration of these partitions by sum is given by A325765.
%e A325764 The sequence of terms together with their prime indices begins:
%e A325764      1: {}
%e A325764      2: {1}
%e A325764      4: {1,1}
%e A325764      6: {1,2}
%e A325764      8: {1,1,1}
%e A325764     16: {1,1,1,1}
%e A325764     18: {1,2,2}
%e A325764     20: {1,1,3}
%e A325764     32: {1,1,1,1,1}
%e A325764     54: {1,2,2,2}
%e A325764     56: {1,1,1,4}
%e A325764     64: {1,1,1,1,1,1}
%e A325764    100: {1,1,3,3}
%e A325764    128: {1,1,1,1,1,1,1}
%e A325764    162: {1,2,2,2,2}
%e A325764    176: {1,1,1,1,5}
%e A325764    256: {1,1,1,1,1,1,1,1}
%e A325764    392: {1,1,1,4,4}
%e A325764    416: {1,1,1,1,1,6}
%e A325764    486: {1,2,2,2,2,2}
%e A325764    500: {1,1,3,3,3}
%e A325764    512: {1,1,1,1,1,1,1,1,1}
%t A325764 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325764 Select[Range[1000],UnsameQ@@Total/@Union[ReplaceList[primeMS[#],{___,s__,___}:>{s}]]&&Range[Total[primeMS[#]]]==Union[ReplaceList[primeMS[#],{___,s__,___}:>Plus[s]]]&]
%Y A325764 Cf. A002033, A056239, A103295, A103300, A112798, A143823, A169942, A325676, A325685, A325763, A325765, A325769, A325770.
%K A325764 nonn
%O A325764 1,2
%A A325764 _Gus Wiseman_, May 20 2019