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A342193
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Numbers with no prime index dividing all the other prime indices.
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18
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1, 15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 91, 93, 95, 99, 105, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 195, 201, 203, 205, 207, 209, 215, 217, 219, 221, 225, 231, 245, 247, 249, 253, 255, 265, 275, 279, 285, 287, 291, 295, 297, 299
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OFFSET
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1,2
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COMMENTS
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Alternative name: 1 and numbers with smallest prime index not dividing all the other prime indices.
First differs from A339562 in having 45.
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 integer partitions with smallest part not dividing all the others (counted by A338470). 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: {} 105: {2,3,4} 201: {2,19}
15: {2,3} 119: {4,7} 203: {4,10}
33: {2,5} 123: {2,13} 205: {3,13}
35: {3,4} 135: {2,2,2,3} 207: {2,2,9}
45: {2,2,3} 141: {2,15} 209: {5,8}
51: {2,7} 143: {5,6} 215: {3,14}
55: {3,5} 145: {3,10} 217: {4,11}
69: {2,9} 153: {2,2,7} 219: {2,21}
75: {2,3,3} 155: {3,11} 221: {6,7}
77: {4,5} 161: {4,9} 225: {2,2,3,3}
85: {3,7} 165: {2,3,5} 231: {2,4,5}
91: {4,6} 175: {3,3,4} 245: {3,4,4}
93: {2,11} 177: {2,17} 247: {6,8}
95: {3,8} 187: {5,7} 249: {2,23}
99: {2,2,5} 195: {2,3,6} 253: {5,9}
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MATHEMATICA
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Select[Range[100], #==1||With[{p=PrimePi/@First/@FactorInteger[#]}, !And@@IntegerQ/@(p/Min@@p)]&]
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CROSSREFS
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The case with maximum prime index not divisible by all others is A343338.
The case with maximum prime index divisible by all others is A343339.
A000070 counts partitions with a selected part.
A001221 counts distinct prime factors.
A299702 lists Heinz numbers of knapsack partitions.
A339564 counts factorizations with a selected factor.
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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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