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A363727
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Numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.
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26
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2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199
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OFFSET
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1,1
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COMMENTS
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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.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
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LINKS
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FORMULA
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EXAMPLE
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The terms together with their prime indices begin:
2: {1} 29: {10} 79: {22}
3: {2} 31: {11} 81: {2,2,2,2}
4: {1,1} 32: {1,1,1,1,1} 83: {23}
5: {3} 37: {12} 89: {24}
7: {4} 41: {13} 90: {1,2,2,3}
8: {1,1,1} 43: {14} 97: {25}
9: {2,2} 47: {15} 101: {26}
11: {5} 49: {4,4} 103: {27}
13: {6} 53: {16} 107: {28}
16: {1,1,1,1} 59: {17} 109: {29}
17: {7} 61: {18} 113: {30}
19: {8} 64: {1,1,1,1,1,1} 121: {5,5}
23: {9} 67: {19} 125: {3,3,3}
25: {3,3} 71: {20} 127: {31}
27: {2,2,2} 73: {21} 128: {1,1,1,1,1,1,1}
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MATHEMATICA
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prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Select[Range[100], {Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]
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CROSSREFS
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A360005 gives twice the median of prime indices.
Just two statistics:
- (median) = (mode): counted by A363740.
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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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