A344297 - OEIS

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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

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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.

LINKS

Table of n, a(n) for n=1..54.

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.

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

Sequence in context: A242661 A336226 A339658 * A344292 A356823 A345359

Adjacent sequences: A344294 A344295 A344296 * A344298 A344299 A344300

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 16 2021

STATUS

approved