A325764 - OEIS

%I #4 May 21 2019 22:05:20

%S 1,2,4,6,8,16,18,20,32,54,56,64,100,128,162,176,256,392,416,486,500,

%T 512,1024,1088,1458,1936,2048,2432,2500,2744,4096,4374,5408,5888,8192,

%U 12500,13122,14848,16384,18496,19208,21296,31744,32768,39366,46208,62500,65536

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

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%C The enumeration of these partitions by sum is given by A325765.

%e The sequence of terms together with their prime indices begins:

%e      1: {}

%e      2: {1}

%e      4: {1,1}

%e      6: {1,2}

%e      8: {1,1,1}

%e     16: {1,1,1,1}

%e     18: {1,2,2}

%e     20: {1,1,3}

%e     32: {1,1,1,1,1}

%e     54: {1,2,2,2}

%e     56: {1,1,1,4}

%e     64: {1,1,1,1,1,1}

%e    100: {1,1,3,3}

%e    128: {1,1,1,1,1,1,1}

%e    162: {1,2,2,2,2}

%e    176: {1,1,1,1,5}

%e    256: {1,1,1,1,1,1,1,1}

%e    392: {1,1,1,4,4}

%e    416: {1,1,1,1,1,6}

%e    486: {1,2,2,2,2,2}

%e    500: {1,1,3,3,3}

%e    512: {1,1,1,1,1,1,1,1,1}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[1000],UnsameQ@@Total/@Union[ReplaceList[primeMS[#],{___,s__,___}:>{s}]]&&Range[Total[primeMS[#]]]==Union[ReplaceList[primeMS[#],{___,s__,___}:>Plus[s]]]&]

%Y Cf. A002033, A056239, A103295, A103300, A112798, A143823, A169942, A325676, A325685, A325763, A325765, A325769, A325770.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 20 2019