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A350845
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Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.
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6
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6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 144, 147, 150, 156, 162, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258, 260, 264, 266, 270
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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), so these are numbers with at least two adjacent prime indices of quotient 1/2.
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LINKS
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EXAMPLE
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The terms and corresponding partitions begin:
6: (2,1)
12: (2,1,1)
18: (2,2,1)
21: (4,2)
24: (2,1,1,1)
30: (3,2,1)
36: (2,2,1,1)
42: (4,2,1)
48: (2,1,1,1,1)
54: (2,2,2,1)
60: (3,2,1,1)
63: (4,2,2)
65: (6,3)
66: (5,2,1)
72: (2,2,1,1,1)
78: (6,2,1)
84: (4,2,1,1)
90: (3,2,2,1)
96: (2,1,1,1,1,1)
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MATHEMATICA
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primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], MemberQ[Divide@@@Partition[primeptn[#], 2, 1], 2]&]
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CROSSREFS
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The strict complement is counted by A350840.
These partitions are counted by A350846.
A000045 = sets containing n with all differences > 2.
A325160 ranks strict partitions with no successions, counted by A003114.
Cf. A000929, A001105, A018819, A045690, A045691, A094537, A154402, A319613, A323093, A337135, A342094, A342095, A342098, A342191.
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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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