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A365830
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Heinz numbers of incomplete integer partitions, meaning not every number from 0 to A056239(n) is the sum of some submultiset.
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22
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3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89
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OFFSET
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1,1
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COMMENTS
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First differs from A325798 in lacking 156.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The complement (complete partitions) is A325781.
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
3: {2}
5: {3}
7: {4}
9: {2,2}
10: {1,3}
11: {5}
13: {6}
14: {1,4}
15: {2,3}
17: {7}
19: {8}
21: {2,4}
22: {1,5}
23: {9}
25: {3,3}
26: {1,6}
27: {2,2,2}
28: {1,1,4}
For example, the submultisets of (1,1,2,6) (right column) and their sums (left column) are:
0: ()
1: (1)
2: (2) or (11)
3: (12)
4: (112)
6: (6)
7: (16)
8: (26) or (116)
9: (126)
10: (1126)
But 5 is missing, so 156 is in the sequence.
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MATHEMATICA
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prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Select[Range[100], Length[nmz[prix[#]]]>0&]
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CROSSREFS
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For prime indices instead of sums we have A080259, complement of A055932.
A299701 counts distinct subset-sums of prime indices.
A365918 counts distinct non-subset-sums of partitions, strict A365922.
A365923 counts partitions by distinct non-subset-sums, strict A365545.
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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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