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A325040
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Heinz numbers of integer partitions with the same product of parts as their conjugate.
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16
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1, 2, 6, 9, 20, 30, 49, 56, 70, 75, 81, 84, 90, 125, 176, 182, 210, 264, 350, 416, 441, 532, 540, 546, 624, 660, 735, 910, 1088, 1100, 1260, 1378, 1386, 1443, 1520, 1560, 1624, 1632, 1715, 2100, 2310, 2401, 2405, 2432, 2489, 2600, 3024, 3267, 3276, 3648, 3744
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OFFSET
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1,2
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COMMENTS
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For example, 182 is the Heinz number of (6,4,1) with product 24 and conjugate (3,2,2,2,1,1) with product also 24.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k).
The enumeration of these partitions by sum is given by A325039.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {}
2: {1}
6: {1,2}
9: {2,2}
20: {1,1,3}
30: {1,2,3}
49: {4,4}
56: {1,1,1,4}
70: {1,3,4}
75: {2,3,3}
81: {2,2,2,2}
84: {1,1,2,4}
90: {1,2,2,3}
125: {3,3,3}
176: {1,1,1,1,5}
182: {1,4,6}
210: {1,2,3,4}
264: {1,1,1,2,5}
350: {1,3,3,4}
416: {1,1,1,1,1,6}
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MATHEMATICA
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priptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Select[Range[100], Times@@priptn[#]==Times@@conj[priptn[#]]&]
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CROSSREFS
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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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