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A344297
Heinz numbers of integer partitions of even numbers with no part greater than 3.
10
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
1,2
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.
FORMULA
Intersection of A051037 and A300061.
EXAMPLE
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}
MATHEMATICA
Select[Range[1000], EvenQ[Total[Cases[FactorInteger[#], {p_, k_}:>k*PrimePi[p]]]]&&Max@@First/@FactorInteger[#]<=Prime[3]&]
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.
Sequence in context: A242661 A336226 A339658 * A344292 A356823 A345359
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 16 2021
STATUS
approved