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Heinz numbers of integer partitions with no part greater than the number of ones.
3

%I #4 May 18 2019 22:46:58

%S 1,2,4,8,12,16,24,32,36,40,48,64,72,80,96,108,112,120,128,144,160,192,

%T 200,216,224,240,256,288,320,324,336,352,360,384,400,432,448,480,512,

%U 560,576,600,640,648,672,704,720,768,784,800,832,864,896,960,972,1000

%N Heinz numbers of integer partitions with no part greater than the number of ones.

%C After 1 and 2, first differs from A322136 in having 200.

%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 A002865.

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

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

%e 36: {1,1,2,2}

%e 40: {1,1,1,3}

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

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

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

%e 80: {1,1,1,1,3}

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

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

%e 112: {1,1,1,1,4}

%e 120: {1,1,1,2,3}

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

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

%t Select[Range[100],#==1||EvenQ[#]&&PrimePi[FactorInteger[#][[-1,1]]]<=FactorInteger[#][[1,2]]&]

%Y Cf. A001222, A002865, A007814, A056239, A061395, A093641, A109298, A110295, A112798, A118914, A325761, A325763.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 18 2019