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Numbers n whose prime indices cover an initial interval of positive integers and include all prime exponents of n.
5

%I #6 May 19 2019 06:17:49

%S 1,2,6,12,18,30,36,60,90,120,150,180,210,270,300,360,420,450,540,600,

%T 630,750,840,900,1050,1080,1260,1350,1470,1500,1680,1800,1890,2100,

%U 2250,2310,2520,2700,2940,3000,3150,3780,4200,4410,4500,4620,5040,5250,5400

%N Numbers n whose prime indices cover an initial interval of positive integers and include all prime exponents of n.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions covering an initial interval of positive integers and containing all of their distinct multiplicities. The enumeration of these partitions by sum is given by A325707.

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

%e 1: {}

%e 2: {1}

%e 6: {1,2}

%e 12: {1,1,2}

%e 18: {1,2,2}

%e 30: {1,2,3}

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

%e 60: {1,1,2,3}

%e 90: {1,2,2,3}

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

%e 150: {1,2,3,3}

%e 180: {1,1,2,2,3}

%e 210: {1,2,3,4}

%e 270: {1,2,2,2,3}

%e 300: {1,1,2,3,3}

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

%e 420: {1,1,2,3,4}

%e 450: {1,2,2,3,3}

%e 540: {1,1,2,2,2,3}

%e 600: {1,1,1,2,3,3}

%t Select[Range[1000],#==1||Range[PrimeNu[#]]==PrimePi/@First/@FactorInteger[#]&&SubsetQ[PrimePi/@First/@FactorInteger[#],Last/@FactorInteger[#]]&]

%Y Cf. A055932, A109297, A114639, A114640, A181819, A290689, A290822, A325705, A325706, A325707.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 18 2019