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Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.
4

%I #5 Feb 19 2023 20:51:57

%S 1,2,6,30,42,49,60,66,70,78,84,90,102,105,114,120,126,132,138,140,150,

%T 154,156,168,174,186,198,204,210,222,228,234,246,258,264,270,276,280,

%U 282,286,294,306,308,312,315,318,330,342,348,350,354,366,372,378,385

%N Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

%C A number's (unordered) prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

%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).

%e The terms together with their prime indices begin:

%e 1: {}

%e 2: {1}

%e 6: {1,2}

%e 30: {1,2,3}

%e 42: {1,2,4}

%e 49: {4,4}

%e 60: {1,1,2,3}

%e 66: {1,2,5}

%e 70: {1,3,4}

%e 78: {1,2,6}

%e 84: {1,1,2,4}

%e 90: {1,2,2,3}

%e For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with median 1. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with median 1/2. So 2760 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],Median[Length/@Split[prix[#]]] == Median[Differences[Prepend[prix[#],0]]]&]

%Y For distinct prime indices instead of 0-prepended differences: A360453.

%Y For mean instead of median we have A360680.

%Y A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.

%Y A124010 gives prime signature, sorted A118914, mean A088529/A088530.

%Y A325347 = partitions w/ integer median, strict A359907, complement A307683.

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

%Y Multisets with integer median:

%Y - For divisors (A063655) we have A139711, complement A139710.

%Y - For prime indices (A360005) we have A359908, complement A359912.

%Y - For distinct prime indices (A360457) we have A360550, complement A360551.

%Y - For distinct prime factors (A360458) we have A360552, complement A100367.

%Y - For prime factors (A360459) we have A359913, complement A072978.

%Y - For prime multiplicities (A360460) we have A360553, complement A360554.

%Y - For 0-prepended differences (A360555) we have A360556, complement A360557.

%Y Cf. A000975, A026424, A316413, A340610, A359904, A360006, A360248, A360558, A360687, A360688.

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 19 2023