proposed

approved

editing

proposed

Cf. ~`A135342, A327473, `A327476, ~`A348551, A359894, `A359897, A359899, `A359900, ~`A359909, A359910, `A359912, `A360006, A360007, `~A360008, A360009.

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

A088529/A088530 gives mean of prime signature A124010.

Cf. A135342 subs_ne_dstnct_means, A316313 meansack, A327473, A327476 h_ptns_wo_mean, A327478 bpe_w_mean, A348551 h_nonint_mean, A359894, A359897 strptns_mean_eq_medn, A359898 strptns_mean_neq_medn, A359899 strnptns_oddlen_mean_eq_medn, A359900 strptns_oddlen_mean_neq_medn, A359903 prix_prisig_eq_mean, A359904 prifacs_prisig_eq_mean, A359905 prix_and_prisig_int_mean, A359906 ptns_int_mean_int_medn, A359909 facs_mean_eq_medn, A359910 facs_oddlen_mean_eq_medn, A359912 prix_nonint_medn, A360006 prix_medn_mins, A360007 prix_medn_sortmins, A360008 prix_mean_sortmins, A360009 prix_int_mean_int_medn.

Cf. ~`A135342, A327473, `A327476, ~`A348551, A359894, `A359897, A359899, `A359900, ~`A359909, A359910, `A359912, `A360006, A360007, `~A360008, A360009.

allocated for Gus WisemanNumbers with odd bigomega 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

1,1

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).

Intersection of A026424 and A359889.

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.

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

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

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.

A088529/A088530 gives mean of prime signature A124010.

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. A135342 subs_ne_dstnct_means, A316313 meansack, A327473, A327476 h_ptns_wo_mean, A327478 bpe_w_mean, A348551 h_nonint_mean, A359894, A359897 strptns_mean_eq_medn, A359898 strptns_mean_neq_medn, A359899 strnptns_oddlen_mean_eq_medn, A359900 strptns_oddlen_mean_neq_medn, A359903 prix_prisig_eq_mean, A359904 prifacs_prisig_eq_mean, A359905 prix_and_prisig_int_mean, A359906 ptns_int_mean_int_medn, A359909 facs_mean_eq_medn, A359910 facs_oddlen_mean_eq_medn, A359912 prix_nonint_medn, A360006 prix_medn_mins, A360007 prix_medn_sortmins, A360008 prix_mean_sortmins, A360009 prix_int_mean_int_medn.

allocated

nonn

Gus Wiseman, Jan 22 2023

approved

editing

allocated for Gus Wiseman

allocated

approved