cp's OEIS Frontend

A359891 Members of A026424 (numbers with an odd number of prime factors) whose prime indices have the same mean as median.

%I A359891 #6 Jan 23 2023 09:10:56
%S A359891 2,3,5,7,8,11,13,17,19,23,27,29,30,31,32,37,41,43,47,53,59,61,67,71,
%T A359891 73,79,83,89,97,101,103,105,107,109,110,113,125,127,128,131,137,139,
%U A359891 149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233
%N A359891 Members of A026424 (numbers with an odd number of prime factors) whose prime indices have the same mean as median.
%C A359891 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 A359891 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 A359891 Intersection of A026424 and A359889.
%e A359891 The terms together with their prime indices begin:
%e A359891    2: {1}
%e A359891    3: {2}
%e A359891    5: {3}
%e A359891    7: {4}
%e A359891    8: {1,1,1}
%e A359891   11: {5}
%e A359891   13: {6}
%e A359891   17: {7}
%e A359891   19: {8}
%e A359891   23: {9}
%e A359891   27: {2,2,2}
%e A359891   29: {10}
%e A359891   30: {1,2,3}
%e A359891   31: {11}
%e A359891   32: {1,1,1,1,1}
%e A359891 For example, the prime indices of 180 are {1,1,2,2,3}, with mean 9/5 and median 2, so 180 is not in the sequence.
%t A359891 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A359891 Select[Range[100],OddQ[PrimeOmega[#]]&&Mean[prix[#]]==Median[prix[#]]&]
%Y A359891 A subset of A026424 = numbers with odd bigomega.
%Y A359891 The LHS (mean of prime indices) is A326567/A326568.
%Y A359891 This is the odd-length case of A359889, complement A359890.
%Y A359891 The complement is A359892.
%Y A359891 These partitions are counted by A359895, any-length A240219.
%Y A359891 The RHS (median of prime indices) is A360005/2.
%Y A359891 A058398 counts partitions by mean, see also A008284, A327482.
%Y A359891 A112798 lists prime indices, length A001222, sum A056239.
%Y A359891 A316413 lists numbers whose prime indices have integer mean.
%Y A359891 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A359891 A359908 lists numbers whose prime indices have integer median.
%Y A359891 Cf. A327473, A359894, A359899, A359910, A360007, A360009.
%K A359891 nonn
%O A359891 1,1
%A A359891 _Gus Wiseman_, Jan 22 2023