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

1, 3, 4, 9, 10, 12, 16, 25, 27, 30, 36, 40, 48, 64, 75, 81, 90, 100, 108, 120, 144, 160, 192, 225, 243, 250, 256, 270, 300, 324, 360, 400, 432, 480, 576, 625, 640, 675, 729, 750, 768, 810, 900, 972, 1000, 1024, 1080, 1200, 1296, 1440, 1600, 1728, 1875, 1920

Offset: 1

Author

Gus Wiseman, May 16 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: {}                 81: {2,2,2,2}
       3: {2}                90: {1,2,2,3}
       4: {1,1}             100: {1,1,3,3}
       9: {2,2}             108: {1,1,2,2,2}
      10: {1,3}             120: {1,1,1,2,3}
      12: {1,1,2}           144: {1,1,1,1,2,2}
      16: {1,1,1,1}         160: {1,1,1,1,1,3}
      25: {3,3}             192: {1,1,1,1,1,1,2}
      27: {2,2,2}           225: {2,2,3,3}
      30: {1,2,3}           243: {2,2,2,2,2}
      36: {1,1,2,2}         250: {1,3,3,3}
      40: {1,1,1,3}         256: {1,1,1,1,1,1,1,1}
      48: {1,1,1,1,2}       270: {1,2,2,2,3}
      64: {1,1,1,1,1,1}     300: {1,1,2,3,3}
      75: {2,3,3}           324: {1,1,2,2,2,2}
		

Crossrefs

These partitions are counted by A007980.

Including partitions of odd numbers gives A051037 (complement: A279622).

Allowing parts > 3 gives A300061.

A001358 lists semiprimes.

A035363 counts partitions whose length is half their sum, ranked by A340387.

A056239 adds up prime indices, row sums of A112798.

Cf. A001399, A002620, A030229, A080193, A244990, A261144, A266755, A344291, A344293, A344294.

Programs

Mathematica

Select[Range[1000],EvenQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]&&Max@@First/@FactorInteger[#]<=Prime[3]&]

Formula

Intersection of A051037 and A300061.