cp's OEIS Frontend

A363727 Numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

%I A363727 #5 Jun 24 2023 07:56:17
%S A363727 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,
%T A363727 61,64,67,71,73,79,81,83,89,90,97,101,103,107,109,113,121,125,127,128,
%U A363727 131,137,139,149,151,157,163,167,169,173,179,181,191,193,197,199
%N A363727 Numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.
%C A363727 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 A363727 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 A363727 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 A363727 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 A363727 The terms together with their prime indices begin:
%e A363727      2: {1}          29: {10}              79: {22}
%e A363727      3: {2}          31: {11}              81: {2,2,2,2}
%e A363727      4: {1,1}        32: {1,1,1,1,1}       83: {23}
%e A363727      5: {3}          37: {12}              89: {24}
%e A363727      7: {4}          41: {13}              90: {1,2,2,3}
%e A363727      8: {1,1,1}      43: {14}              97: {25}
%e A363727      9: {2,2}        47: {15}             101: {26}
%e A363727     11: {5}          49: {4,4}            103: {27}
%e A363727     13: {6}          53: {16}             107: {28}
%e A363727     16: {1,1,1,1}    59: {17}             109: {29}
%e A363727     17: {7}          61: {18}             113: {30}
%e A363727     19: {8}          64: {1,1,1,1,1,1}    121: {5,5}
%e A363727     23: {9}          67: {19}             125: {3,3,3}
%e A363727     25: {3,3}        71: {20}             127: {31}
%e A363727     27: {2,2,2}      73: {21}             128: {1,1,1,1,1,1,1}
%t A363727 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363727 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363727 Select[Range[100],{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]
%Y A363727 These partitions are counted by A363719, factorizations A363741.
%Y A363727 For unequal instead of equal we have A363730, counted by A363720.
%Y A363727 Excluding primes gives A363722.
%Y A363727 Excluding prime-powers gives A363729, counted by A363728.
%Y A363727 A112798 lists prime indices, length A001222, sum A056239.
%Y A363727 A326567/A326568 gives mean of prime indices.
%Y A363727 A356862 ranks partitions with a unique mode, counted by A362608.
%Y A363727 A359178 ranks partitions with multiple modes, counted by A362610.
%Y A363727 A360005 gives twice the median of prime indices.
%Y A363727 A362611 counts modes in prime indices, triangle A362614.
%Y A363727 A362613 counts co-modes in prime indices, triangle A362615.
%Y A363727 A363486 gives least mode in prime indices, A363487 greatest.
%Y A363727 Just two statistics:
%Y A363727 - (mean) = (median): A359889, counted by A240219.
%Y A363727 - (mean) != (median): A359890, counted by A359894.
%Y A363727 - (mean) = (mode): counted by A363723, see A363724, A363731.
%Y A363727 - (median) = (mode): counted by A363740.
%Y A363727 Cf. A215366, A327473, A327476, A359893, A359908, A360009, A360248, A360550, A363721, A363725.
%K A363727 nonn
%O A363727 1,1
%A A363727 _Gus Wiseman_, Jun 23 2023