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A339562
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Squarefree numbers with no prime index dividing all the other prime indices.
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16
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1, 15, 33, 35, 51, 55, 69, 77, 85, 91, 93, 95, 105, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 195, 201, 203, 205, 209, 215, 217, 219, 221, 231, 247, 249, 253, 255, 265, 285, 287, 291, 295, 299, 301, 309, 323, 327, 329, 335, 341, 345, 355, 357, 377, 381
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OFFSET
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1,2
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COMMENTS
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First differs from A342193 in lacking 45.
Alternative name: 1 and squarefree numbers with smallest prime index not dividing all the other prime indices.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also 1 and Heinz numbers of strict integer partitions with smallest part not dividing all the others (counted by A341450). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {} 141: {2,15} 219: {2,21}
15: {2,3} 143: {5,6} 221: {6,7}
33: {2,5} 145: {3,10} 231: {2,4,5}
35: {3,4} 155: {3,11} 247: {6,8}
51: {2,7} 161: {4,9} 249: {2,23}
55: {3,5} 165: {2,3,5} 253: {5,9}
69: {2,9} 177: {2,17} 255: {2,3,7}
77: {4,5} 187: {5,7} 265: {3,16}
85: {3,7} 195: {2,3,6} 285: {2,3,8}
91: {4,6} 201: {2,19} 287: {4,13}
93: {2,11} 203: {4,10} 291: {2,25}
95: {3,8} 205: {3,13} 295: {3,17}
105: {2,3,4} 209: {5,8} 299: {6,9}
119: {4,7} 215: {3,14} 301: {4,14}
123: {2,13} 217: {4,11} 309: {2,27}
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MATHEMATICA
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Select[Range[100], #==1||SquareFreeQ[#]&&With[{p=PrimePi/@First/@FactorInteger[#]}, !And@@IntegerQ/@(p/Min@@p)]&]
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CROSSREFS
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The squarefree complement is A339563.
These partitions are counted by A341450.
The not necessarily squarefree version is A342193.
A000070 counts partitions with a selected part.
A001221 counts distinct prime factors.
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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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