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A350838
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Heinz numbers of partitions with no adjacent parts of quotient 2.
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9
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1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83
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OFFSET
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1,2
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COMMENTS
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Differs from A320340 in having 105: (4,3,2), 315: (4,3,2,2), 455: (6,4,3), etc.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with no adjacent prime indices of quotient 1/2.
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LINKS
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EXAMPLE
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The terms and their prime indices begin:
1: {} 19: {8} 38: {1,8}
2: {1} 20: {1,1,3} 39: {2,6}
3: {2} 22: {1,5} 40: {1,1,1,3}
4: {1,1} 23: {9} 41: {13}
5: {3} 25: {3,3} 43: {14}
7: {4} 26: {1,6} 44: {1,1,5}
8: {1,1,1} 27: {2,2,2} 45: {2,2,3}
9: {2,2} 28: {1,1,4} 46: {1,9}
10: {1,3} 29: {10} 47: {15}
11: {5} 31: {11} 49: {4,4}
13: {6} 32: {1,1,1,1,1} 50: {1,3,3}
14: {1,4} 33: {2,5} 51: {2,7}
15: {2,3} 34: {1,7} 52: {1,1,6}
16: {1,1,1,1} 35: {3,4} 53: {16}
17: {7} 37: {12} 55: {3,5}
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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], And@@Table[FreeQ[Divide@@@Partition[primeptn[#], 2, 1], 2], {i, 2, PrimeOmega[#]}]&]
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CROSSREFS
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The sets version (subsets of prescribed maximum) is counted by A045691.
These partitions are counted by A350837.
The strict case is counted by A350840.
A000045 = sets containing n with all differences > 2.
Cf. A000302, A001105, A003000, A018819, A094537, A120641, A154402, A319613, A323093, A337135, A342097, A342095.
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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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