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A341446
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Heinz numbers of integer partitions whose only odd part is the smallest.
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15
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2, 5, 6, 11, 14, 17, 18, 23, 26, 31, 35, 38, 41, 42, 47, 54, 58, 59, 65, 67, 73, 74, 78, 83, 86, 95, 97, 98, 103, 106, 109, 114, 122, 126, 127, 137, 142, 143, 145, 149, 157, 158, 162, 167, 174, 178, 179, 182, 185, 191, 197, 202, 209, 211, 214, 215, 222, 226
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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 whose only odd prime index (counting multiplicity) is the smallest.
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LINKS
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FORMULA
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EXAMPLE
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The sequence of partitions together with their Heinz numbers begins:
2: (1) 54: (2,2,2,1) 109: (29)
5: (3) 58: (10,1) 114: (8,2,1)
6: (2,1) 59: (17) 122: (18,1)
11: (5) 65: (6,3) 126: (4,2,2,1)
14: (4,1) 67: (19) 127: (31)
17: (7) 73: (21) 137: (33)
18: (2,2,1) 74: (12,1) 142: (20,1)
23: (9) 78: (6,2,1) 143: (6,5)
26: (6,1) 83: (23) 145: (10,3)
31: (11) 86: (14,1) 149: (35)
35: (4,3) 95: (8,3) 157: (37)
38: (8,1) 97: (25) 158: (22,1)
41: (13) 98: (4,4,1) 162: (2,2,2,2,1)
42: (4,2,1) 103: (27) 167: (39)
47: (15) 106: (16,1) 174: (10,2,1)
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 100], OddQ[First[primeMS[#]]]&&And@@EvenQ[Rest[primeMS[#]]]&]
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CROSSREFS
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These partitions are counted by A035363 (shifted left once).
Terms of A340932 can be factored into elements of this sequence.
A026804 counts partitions whose smallest part is odd.
A032742 selects largest proper divisor.
A055396 selects smallest prime index.
A061395 selects largest prime index.
A066207 lists numbers with all even prime indices.
A066208 lists numbers with all odd prime indices.
A112798 lists the prime indices of each positive integer.
A244991 lists numbers whose greatest prime index is odd.
A340932 lists numbers whose smallest prime index is odd.
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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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