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%I #5 Jun 24 2023 07:56:08
%S 4,8,9,16,25,27,32,49,64,81,90,121,125,128,169,243,256,270,289,343,
%T 361,512,525,529,550,625,729,756,810,841,961,1024,1331,1369,1666,1681,
%U 1849,1911,1950,2048,2187,2197,2209,2268,2401,2430,2625,2695,2700,2750,2809
%N Nonprime numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.
%C 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.
%C 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}.
%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%F Complement of A000040 in A363727.
%F Assuming there is a unique mode, we have A326567(a(n))/A326568(a(n)) = A360005(a(n))/2 = A363486(a(n)) = A363487(a(n)).
%e The terms together with their prime indices begin:
%e 4: {1,1}
%e 8: {1,1,1}
%e 9: {2,2}
%e 16: {1,1,1,1}
%e 25: {3,3}
%e 27: {2,2,2}
%e 32: {1,1,1,1,1}
%e 49: {4,4}
%e 64: {1,1,1,1,1,1}
%e 81: {2,2,2,2}
%e 90: {1,2,2,3}
%e 121: {5,5}
%e 125: {3,3,3}
%e 128: {1,1,1,1,1,1,1}
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t Select[Range[100],!PrimeQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]
%Y These partitions are counted by A363719 - 1 for n > 0.
%Y Including primes gives A363727, counted by A363719.
%Y For prime powers instead of just primes we have A363729, counted by A363728.
%Y For unequal instead of equal we have A363730, counted by A363720.
%Y A112798 lists prime indices, length A001222, sum A056239.
%Y A326567/A326568 gives mean of prime indices.
%Y A356862 ranks partitions with a unique mode, counted by A362608.
%Y A359178 ranks partitions with multiple modes, counted by A362610.
%Y A360005 gives twice the median of prime indices.
%Y A362611 counts modes in prime indices, triangle A362614.
%Y A362613 counts co-modes in prime indices, triangle A362615.
%Y A363486 gives least mode in prime indices, A363487 greatest.
%Y Just two statistics:
%Y - (mean) = (median): A359889, counted by A240219.
%Y - (mean) != (median): A359890, counted by A359894.
%Y - (mean) = (mode): counted by A363723, see A363724, A363731.
%Y - (median) = (mode): counted by A363740.
%Y Cf. A000961, A327473, A327476, A359893, A359908, A360009, A360550, A363721, A363725, A363741.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 24 2023
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