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