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A363722
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Nonprime numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.
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5
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4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 90, 121, 125, 128, 169, 243, 256, 270, 289, 343, 361, 512, 525, 529, 550, 625, 729, 756, 810, 841, 961, 1024, 1331, 1369, 1666, 1681, 1849, 1911, 1950, 2048, 2187, 2197, 2209, 2268, 2401, 2430, 2625, 2695, 2700, 2750, 2809
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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:
4: {1,1}
8: {1,1,1}
9: {2,2}
16: {1,1,1,1}
25: {3,3}
27: {2,2,2}
32: {1,1,1,1,1}
49: {4,4}
64: {1,1,1,1,1,1}
81: {2,2,2,2}
90: {1,2,2,3}
121: {5,5}
125: {3,3,3}
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], !PrimeQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]
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CROSSREFS
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These partitions are counted by A363719 - 1 for n > 0.
For prime powers instead of just primes we have A363729, counted by A363728.
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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