A359891 - OEIS

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A359891

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

2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 30, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 105, 107, 109, 110, 113, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

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OFFSET

1,1

COMMENTS

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.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

LINKS

Table of n, a(n) for n=1..59.

FORMULA

Intersection of A026424 and A359889.

EXAMPLE

The terms together with their prime indices begin:

2: {1}

3: {2}

5: {3}

7: {4}

8: {1,1,1}

11: {5}

13: {6}

17: {7}

19: {8}

23: {9}

27: {2,2,2}

29: {10}

30: {1,2,3}

31: {11}

32: {1,1,1,1,1}

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.

MATHEMATICA

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

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

CROSSREFS

A subset of A026424 = numbers with odd bigomega.

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

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

The complement is A359892.

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

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

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

A112798 lists prime indices, length A001222, sum A056239.

A316413 lists numbers whose prime indices have integer mean.

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

A359908 lists numbers whose prime indices have integer median.

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

Sequence in context: A079851 A060634 A279457 * A171561 A246551 A268391

Adjacent sequences: A359888 A359889 A359890 * A359892 A359893 A359894

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 22 2023

STATUS

approved