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A344295
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Heinz numbers of partitions of 2*n with at most n parts, none greater than 3, for some n.
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4
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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Intersection of A300061 (even Heinz weight), A344291 (Omega > half Heinz weight), and A051037 (5-smooth).
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EXAMPLE
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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}
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MATHEMATICA
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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&]
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CROSSREFS
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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.
A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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