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A352486
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Heinz numbers of non-self-conjugate integer partitions.
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24
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3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73
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OFFSET
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1,1
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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). This gives a bijective correspondence between positive integers and integer partitions. The sequence lists all Heinz numbers of partitions whose Heinz number is different from that of their conjugate.
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LINKS
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FORMULA
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EXAMPLE
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The terms together with their prime indices begin:
3: (2)
4: (1,1)
5: (3)
7: (4)
8: (1,1,1)
10: (3,1)
11: (5)
12: (2,1,1)
13: (6)
14: (4,1)
15: (3,2)
16: (1,1,1,1)
17: (7)
18: (2,2,1)
For example, the self-conjugate partition (4,3,3,1) has Heinz number 350, so 350 is not in the sequence.
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
conj[y_]:=If[Length[y0]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Select[Range[100], #!=Times@@Prime/@conj[primeMS[#]]&]
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CROSSREFS
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These partitions are counted by A330644.
These are the positions of nonzero terms in A352491.
A098825 counts permutations by unfixed points.
A325039 counts partitions w/ same product as conjugate, ranked by A325040.
A352523 counts compositions by unfixed points, rank statistic A352513.
Heinz number (rank) and partition:
- A122111 = rank of conjugate partition
Cf. A000720, A026424, A120383, A175508, A195017, A238745, A301987, A304360, A316524, A324846, A350841.
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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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