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Members of A026424 (numbers with an odd number of prime factors) whose prime indices do not have the same mean as median.
7

%I #6 Jan 23 2023 09:11:00

%S 12,18,20,28,42,44,45,48,50,52,63,66,68,70,72,75,76,78,80,92,98,99,

%T 102,108,112,114,116,117,120,124,130,138,147,148,153,154,162,164,165,

%U 168,170,171,172,174,175,176,180,182,186,188,190,192,195,200,207,208

%N Members of A026424 (numbers with an odd number of prime factors) whose prime indices do not 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 A359890.

%e The terms together with their prime indices begin:

%e 12: {1,1,2}

%e 18: {1,2,2}

%e 20: {1,1,3}

%e 28: {1,1,4}

%e 42: {1,2,4}

%e 44: {1,1,5}

%e 45: {2,2,3}

%e 48: {1,1,1,1,2}

%e 50: {1,3,3}

%e 52: {1,1,6}

%e 63: {2,2,4}

%e 66: {1,2,5}

%e 68: {1,1,7}

%e 70: {1,3,4}

%e 72: {1,1,1,2,2}

%e For example, the prime indices of 180 are {1,1,2,2,3}, with mean 9/5 and median 2, so 180 is 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 A359890, complement A359889.

%Y The complement is A359891.

%Y These partitions are counted by A359896, complement A359895.

%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 A359902 counts odd-length partitions by median.

%Y Cf. A240219, A327473, A327476, A348551, A359894, A359898, A359899, A359900, A359911, A359912, A360006-A360009.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 22 2023