cp's OEIS Frontend

A363730 Numbers whose prime indices have different mean, median, and mode.

%I A363730 #5 Jun 24 2023 13:03:02
%S A363730 42,60,66,70,78,84,102,114,130,132,138,140,150,154,156,165,170,174,
%T A363730 180,182,186,190,195,204,220,222,228,230,231,246,255,258,260,266,276,
%U A363730 282,285,286,290,294,308,310,315,318,322,330,340,345,348,354,357,360,364
%N A363730 Numbers whose prime indices have different mean, median, and mode.
%C A363730 If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
%C A363730 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 A363730 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 A363730 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 A363730 All three of A326567(a(n))/A326568(a(n)), A360005(a(n))/2, and A363486(a(n)) = A363487(a(n)) are different.
%e A363730 The prime indices of 180 are {1,1,2,2,3}, with mean 9/5, median 2, modes {1,2}, so 180 is in the sequence.
%e A363730 The prime indices of 108 are {1,1,2,2,2}, with mean 8/5, median 2, modes {2}, so 108 is not in the sequence.
%e A363730 The terms together with their prime indices begin:
%e A363730    42: {1,2,4}
%e A363730    60: {1,1,2,3}
%e A363730    66: {1,2,5}
%e A363730    70: {1,3,4}
%e A363730    78: {1,2,6}
%e A363730    84: {1,1,2,4}
%e A363730   102: {1,2,7}
%e A363730   114: {1,2,8}
%e A363730   130: {1,3,6}
%e A363730   132: {1,1,2,5}
%e A363730   138: {1,2,9}
%e A363730   140: {1,1,3,4}
%e A363730   150: {1,2,3,3}
%t A363730 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363730 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363730 Select[Range[100],{Mean[prix[#]]}!={Median[prix[#]]}!=modes[prix[#]]&]
%Y A363730 These partitions are counted by A363720
%Y A363730 For equal instead of unequal we have A363727, counted by A363719.
%Y A363730 The version for factorizations is A363742, equal A363741.
%Y A363730 A112798 lists prime indices, length A001222, sum A056239.
%Y A363730 A326567/A326568 gives mean of prime indices.
%Y A363730 A356862 ranks partitions with a unique mode, counted by A362608.
%Y A363730 A359178 ranks partitions with multiple modes, counted by A362610.
%Y A363730 A360005 gives twice the median of prime indices.
%Y A363730 A362611 counts modes in prime indices, triangle A362614.
%Y A363730 A362613 counts co-modes in prime indices, triangle A362615.
%Y A363730 A363486 gives least mode in prime indices, A363487 greatest.
%Y A363730 Just two statistics:
%Y A363730 - (mean) = (median): A359889, counted by A240219.
%Y A363730 - (mean) != (median): A359890, counted by A359894.
%Y A363730 - (mean) = (mode): counted by A363723, see A363724, A363731.
%Y A363730 - (median) = (mode): counted by A363740.
%Y A363730 Cf. A000961, A327473, A327476, A359908, A363722, A363725, A363729.
%K A363730 nonn
%O A363730 1,1
%A A363730 _Gus Wiseman_, Jun 24 2023