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Heinz numbers of integer partitions covering an initial interval of positive integers with distinct multiplicities.
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%I #4 May 02 2019 08:52:39

%S 1,2,4,8,12,16,18,24,32,48,54,64,72,96,108,128,144,162,192,256,288,

%T 324,360,384,432,486,512,540,576,600,648,720,768,864,972,1024,1152,

%U 1200,1350,1440,1458,1500,1536,1620,1728,1944,2048,2160,2250,2304,2400,2592

%N Heinz numbers of integer partitions covering an initial interval of positive integers with distinct multiplicities.

%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 A320348.

%F Intersection of normal numbers (A055932) and numbers with distinct prime exponents (A130091).

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

%e 1: {}

%e 2: {1}

%e 4: {1,1}

%e 8: {1,1,1}

%e 12: {1,1,2}

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

%e 18: {1,2,2}

%e 24: {1,1,1,2}

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

%e 48: {1,1,1,1,2}

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

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

%e 72: {1,1,1,2,2}

%e 96: {1,1,1,1,1,2}

%e 108: {1,1,2,2,2}

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

%e 144: {1,1,1,1,2,2}

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

%e 192: {1,1,1,1,1,1,2}

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

%e 288: {1,1,1,1,1,2,2}

%e 324: {1,1,2,2,2,2}

%e 360: {1,1,1,2,2,3}

%e 384: {1,1,1,1,1,1,1,2}

%t normQ[n_Integer]:=n==1||PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]];

%t Select[Range[100],normQ[#]&&UnsameQ@@Last/@FactorInteger[#]&]

%Y Cf. A000837, A047966, A055932, A056239, A098859, A112798, A130091, A317081, A317089, A320348, A325329, A325337, A325369, A325372.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 01 2019