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A344297 Heinz numbers of integer partitions of even numbers with no part greater than 3.

%I A344297 #6 May 20 2021 23:05:44
%S A344297 1,3,4,9,10,12,16,25,27,30,36,40,48,64,75,81,90,100,108,120,144,160,
%T A344297 192,225,243,250,256,270,300,324,360,400,432,480,576,625,640,675,729,
%U A344297 750,768,810,900,972,1000,1024,1080,1200,1296,1440,1600,1728,1875,1920
%N A344297 Heinz numbers of integer partitions of even numbers with no part greater than 3.
%C A344297 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
%F A344297 Intersection of A051037 and A300061.
%e A344297 The sequence of terms together with their prime indices begins:
%e A344297        1: {}                 81: {2,2,2,2}
%e A344297        3: {2}                90: {1,2,2,3}
%e A344297        4: {1,1}             100: {1,1,3,3}
%e A344297        9: {2,2}             108: {1,1,2,2,2}
%e A344297       10: {1,3}             120: {1,1,1,2,3}
%e A344297       12: {1,1,2}           144: {1,1,1,1,2,2}
%e A344297       16: {1,1,1,1}         160: {1,1,1,1,1,3}
%e A344297       25: {3,3}             192: {1,1,1,1,1,1,2}
%e A344297       27: {2,2,2}           225: {2,2,3,3}
%e A344297       30: {1,2,3}           243: {2,2,2,2,2}
%e A344297       36: {1,1,2,2}         250: {1,3,3,3}
%e A344297       40: {1,1,1,3}         256: {1,1,1,1,1,1,1,1}
%e A344297       48: {1,1,1,1,2}       270: {1,2,2,2,3}
%e A344297       64: {1,1,1,1,1,1}     300: {1,1,2,3,3}
%e A344297       75: {2,3,3}           324: {1,1,2,2,2,2}
%t A344297 Select[Range[1000],EvenQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]&&Max@@First/@FactorInteger[#]<=Prime[3]&]
%Y A344297 These partitions are counted by A007980.
%Y A344297 Including partitions of odd numbers gives A051037 (complement: A279622).
%Y A344297 Allowing parts > 3 gives A300061.
%Y A344297 A001358 lists semiprimes.
%Y A344297 A035363 counts partitions whose length is half their sum, ranked by A340387.
%Y A344297 A056239 adds up prime indices, row sums of A112798.
%Y A344297 Cf. A001399, A002620, A030229, A080193, A244990, A261144, A266755, A344291, A344293, A344294.
%K A344297 nonn
%O A344297 1,2
%A A344297 _Gus Wiseman_, May 16 2021