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

1, 3, 9, 10, 25, 27, 30, 75, 81, 90, 100, 225, 243, 250, 270, 300, 625, 675, 729, 750, 810, 900, 1000, 1875, 2025, 2187, 2250, 2430, 2500, 2700, 3000, 5625, 6075, 6250, 6561, 6750, 7290, 7500, 8100, 9000, 10000, 15625, 16875, 18225, 18750, 19683, 20250, 21870

Offset: 1

Author

Gus Wiseman, May 15 2021

Keywords

• nonn

Comments

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.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}
      3: {2}
      9: {2,2}
     10: {1,3}
     25: {3,3}
     27: {2,2,2}
     30: {1,2,3}
     75: {2,3,3}
     81: {2,2,2,2}
     90: {1,2,2,3}
    100: {1,1,3,3}
    225: {2,2,3,3}
    243: {2,2,2,2,2}
    250: {1,3,3,3}
    270: {1,2,2,2,3}
    300: {1,1,2,3,3}
		

Crossrefs

These partitions are counted by A001399.

Allowing any number of parts and sum gives A051037.

Allowing parts > 3 and any length gives A300061.

Not requiring the sum of prime indices to be even gives A344293.

Allowing any number of parts (but still with even sum) gives A344297.

Allowing parts > 3 gives A344413.

A001358 lists semiprimes.

A025065 counts partitions of n with at least n/2 parts, ranked by A344296.

A035363 counts partitions of n of length n/2, ranked by A340387.

A056239 adds up prime indices, row sums of A112798.

A110618 counts partitions of n with at most n/2 parts, ranked by A344291.

A344414 counts partitions of n with all parts >= n/2, ranked by A344296.

Cf. A030229, A080193, A244990, A261144, A266755, A279622, A334433, A344294.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],EvenQ[Total[primeMS[#]]]&&PrimeOmega[#]<=Total[primeMS[#]]/2&&Max@@primeMS[#]<=3&]

Formula

Intersection of A300061 (even Heinz weight), A344291 (Omega > half Heinz weight), and A051037 (5-smooth).