A359891 - OEIS

%I #6 Jan 23 2023 09:10:56

%S 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 73,79,83,89,97,101,103,105,107,109,110,113,125,127,128,131,137,139,

%U 149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233

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

%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 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 Intersection of A026424 and A359889.

%e The terms together with their prime indices begin:

%e    2: {1}

%e    3: {2}

%e    5: {3}

%e    7: {4}

%e    8: {1,1,1}

%e   11: {5}

%e   13: {6}

%e   17: {7}

%e   19: {8}

%e   23: {9}

%e   27: {2,2,2}

%e   29: {10}

%e   30: {1,2,3}

%e   31: {11}

%e   32: {1,1,1,1,1}

%e 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 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],OddQ[PrimeOmega[#]]&&Mean[prix[#]]==Median[prix[#]]&]

%Y A subset of A026424 = numbers with odd bigomega.

%Y The LHS (mean of prime indices) is A326567/A326568.

%Y This is the odd-length case of A359889, complement A359890.

%Y The complement is A359892.

%Y These partitions are counted by A359895, any-length A240219.

%Y The RHS (median of prime indices) is A360005/2.

%Y A058398 counts partitions by mean, see also A008284, A327482.

%Y A112798 lists prime indices, length A001222, sum A056239.

%Y A316413 lists numbers whose prime indices have integer mean.

%Y A359893 and A359901 count partitions by median, odd-length A359902.

%Y A359908 lists numbers whose prime indices have integer median.

%Y Cf. A327473, A359894, A359899, A359910, A360007, A360009.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 22 2023