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A370811
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Numbers such that more than one set can be obtained by choosing a different divisor of each prime index.
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5
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3, 5, 7, 11, 13, 14, 15, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 65, 67, 69, 70, 71, 73, 74, 77, 78, 79, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 105, 106, 107, 109, 111, 113, 114, 115, 117, 119
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OFFSET
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1,1
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COMMENTS
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A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798.
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LINKS
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EXAMPLE
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The prime indices of 70 are {1,3,4}, with choices (1,3,4) and (1,3,2), so 70 is in the sequence.
The terms together with their prime indices begin:
3: {2} 43: {14} 79: {22} 115: {3,9}
5: {3} 46: {1,9} 83: {23} 117: {2,2,6}
7: {4} 47: {15} 85: {3,7} 119: {4,7}
11: {5} 49: {4,4} 86: {1,14} 122: {1,18}
13: {6} 51: {2,7} 87: {2,10} 123: {2,13}
14: {1,4} 53: {16} 89: {24} 127: {31}
15: {2,3} 55: {3,5} 91: {4,6} 129: {2,14}
17: {7} 57: {2,8} 93: {2,11} 130: {1,3,6}
19: {8} 58: {1,10} 94: {1,15} 131: {32}
21: {2,4} 59: {17} 95: {3,8} 133: {4,8}
23: {9} 61: {18} 97: {25} 137: {33}
26: {1,6} 65: {3,6} 101: {26} 138: {1,2,9}
29: {10} 67: {19} 103: {27} 139: {34}
31: {11} 69: {2,9} 105: {2,3,4} 141: {2,15}
33: {2,5} 70: {1,3,4} 106: {1,16} 142: {1,20}
35: {3,4} 71: {20} 107: {28} 143: {5,6}
37: {12} 73: {21} 109: {29} 145: {3,10}
38: {1,8} 74: {1,12} 111: {2,12} 146: {1,21}
39: {2,6} 77: {4,5} 113: {30} 149: {35}
41: {13} 78: {1,2,6} 114: {1,2,8} 151: {36}
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MATHEMATICA
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prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Length[Union[Sort /@ Select[Tuples[Divisors/@prix[#]], UnsameQ@@#&]]]>1&]
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CROSSREFS
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Partitions of this type are counted by A370803.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370814 counts factorizations with choosable divisors, complement A370813.
Cf. A133686, A355529, A355739, A355749, A367771, A367904, A370584, A370592, A370594, A370647, A370808, A370816.
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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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