cp's OEIS Frontend

A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

%I A363729 #5 Jun 24 2023 07:56:13
%S A363729 90,270,525,550,756,810,1666,1911,1950,2268,2430,2625,2695,2700,2750,
%T A363729 5566,6762,6804,6897,7128,7290,8100,8500,9310,9750,10285,10478,11011,
%U A363729 11550,11662,12250,12375,12495,13125,13377,13750,14014,14703,18865,19435,20412,21384
%N A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.
%C A363729 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 A363729 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 A363729 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%e A363729 The prime indices of 6897 are {2,5,5,8}, with mean 5, median 5, and modes {5}, so 6897 is in the sequence.
%e A363729 The terms together with their prime indices begin:
%e A363729      90: {1,2,2,3}
%e A363729     270: {1,2,2,2,3}
%e A363729     525: {2,3,3,4}
%e A363729     550: {1,3,3,5}
%e A363729     756: {1,1,2,2,2,4}
%e A363729     810: {1,2,2,2,2,3}
%e A363729    1666: {1,4,4,7}
%e A363729    1911: {2,4,4,6}
%e A363729    1950: {1,2,3,3,6}
%e A363729    2268: {1,1,2,2,2,2,4}
%e A363729    2430: {1,2,2,2,2,2,3}
%t A363729 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363729 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363729 Select[Range[1000],!PrimePowerQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]
%Y A363729 For just primes instead of prime powers we have A363722.
%Y A363729 Including prime-powers gives A363727, counted by A363719.
%Y A363729 These partitions are counted by A363728.
%Y A363729 For unequal instead of equal we have A363730, counted by A363720.
%Y A363729 A000961 lists the prime powers, complement A024619.
%Y A363729 A112798 lists prime indices, length A001222, sum A056239.
%Y A363729 A326567/A326568 gives mean of prime indices.
%Y A363729 A356862 ranks partitions with a unique mode, counted by A362608.
%Y A363729 A359178 ranks partitions with multiple modes, counted by A362610.
%Y A363729 A360005 gives twice the median of prime indices.
%Y A363729 A362611 counts modes in prime indices, triangle A362614.
%Y A363729 A362613 counts co-modes in prime indices, triangle A362615.
%Y A363729 A363486 gives least mode in prime indices, A363487 greatest.
%Y A363729 Just two statistics:
%Y A363729 - (mean) = (median): A359889, counted by A240219.
%Y A363729 - (mean) != (median): A359890, counted by A359894.
%Y A363729 - (mean) = (mode): counted by A363723, see A363724, A363731.
%Y A363729 - (median) = (mode): counted by A363740.
%Y A363729 Cf. A215366, A327473, A327476, A359893, A359908, A360009, A360248, A360550, A363721, A363725, A363741.
%K A363729 nonn
%O A363729 1,1
%A A363729 _Gus Wiseman_, Jun 24 2023