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A344295 Heinz numbers of partitions of 2*n with at most n parts, none greater than 3, for some n.

%I A344295 #12 May 20 2021 23:05:13
%S A344295 1,3,9,10,25,27,30,75,81,90,100,225,243,250,270,300,625,675,729,750,
%T A344295 810,900,1000,1875,2025,2187,2250,2430,2500,2700,3000,5625,6075,6250,
%U A344295 6561,6750,7290,7500,8100,9000,10000,15625,16875,18225,18750,19683,20250,21870
%N A344295 Heinz numbers of partitions of 2*n with at most n parts, none greater than 3, for some n.
%C A344295 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 A344295 Intersection of A300061 (even Heinz weight), A344291 (Omega > half Heinz weight), and A051037 (5-smooth).
%e A344295 The sequence of terms together with their prime indices begins:
%e A344295       1: {}
%e A344295       3: {2}
%e A344295       9: {2,2}
%e A344295      10: {1,3}
%e A344295      25: {3,3}
%e A344295      27: {2,2,2}
%e A344295      30: {1,2,3}
%e A344295      75: {2,3,3}
%e A344295      81: {2,2,2,2}
%e A344295      90: {1,2,2,3}
%e A344295     100: {1,1,3,3}
%e A344295     225: {2,2,3,3}
%e A344295     243: {2,2,2,2,2}
%e A344295     250: {1,3,3,3}
%e A344295     270: {1,2,2,2,3}
%e A344295     300: {1,1,2,3,3}
%t A344295 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A344295 Select[Range[1000],EvenQ[Total[primeMS[#]]]&&PrimeOmega[#]<=Total[primeMS[#]]/2&&Max@@primeMS[#]<=3&]
%Y A344295 These partitions are counted by A001399.
%Y A344295 Allowing any number of parts and sum gives A051037.
%Y A344295 Allowing parts > 3 and any length gives A300061.
%Y A344295 Not requiring the sum of prime indices to be even gives A344293.
%Y A344295 Allowing any number of parts (but still with even sum) gives A344297.
%Y A344295 Allowing parts > 3 gives A344413.
%Y A344295 A001358 lists semiprimes.
%Y A344295 A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
%Y A344295 A035363 counts partitions of n of length n/2, ranked by A340387.
%Y A344295 A056239 adds up prime indices, row sums of A112798.
%Y A344295 A110618 counts partitions of n with at most n/2 parts, ranked by A344291.
%Y A344295 A344414 counts partitions of n with all parts >= n/2, ranked by A344296.
%Y A344295 Cf. A030229, A080193, A244990, A261144, A266755, A279622, A334433, A344294.
%K A344295 nonn
%O A344295 1,2
%A A344295 _Gus Wiseman_, May 15 2021