cp's OEIS Frontend

A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

%I A360681 #5 Feb 19 2023 20:51:57
%S A360681 1,2,6,30,42,49,60,66,70,78,84,90,102,105,114,120,126,132,138,140,150,
%T A360681 154,156,168,174,186,198,204,210,222,228,234,246,258,264,270,276,280,
%U A360681 282,286,294,306,308,312,315,318,330,342,348,350,354,366,372,378,385
%N A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.
%C A360681 A number's (unordered) prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
%C A360681 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 A360681 The terms together with their prime indices begin:
%e A360681     1: {}
%e A360681     2: {1}
%e A360681     6: {1,2}
%e A360681    30: {1,2,3}
%e A360681    42: {1,2,4}
%e A360681    49: {4,4}
%e A360681    60: {1,1,2,3}
%e A360681    66: {1,2,5}
%e A360681    70: {1,3,4}
%e A360681    78: {1,2,6}
%e A360681    84: {1,1,2,4}
%e A360681    90: {1,2,2,3}
%e A360681 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 A360681 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A360681 Select[Range[100],Median[Length/@Split[prix[#]]] == Median[Differences[Prepend[prix[#],0]]]&]
%Y A360681 For distinct prime indices instead of 0-prepended differences: A360453.
%Y A360681 For mean instead of median we have A360680.
%Y A360681 A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
%Y A360681 A124010 gives prime signature, sorted A118914, mean A088529/A088530.
%Y A360681 A325347 = partitions w/ integer median, strict A359907, complement A307683.
%Y A360681 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A360681 Multisets with integer median:
%Y A360681 - For divisors (A063655) we have A139711, complement A139710.
%Y A360681 - For prime indices (A360005) we have A359908, complement A359912.
%Y A360681 - For distinct prime indices (A360457) we have A360550, complement A360551.
%Y A360681 - For distinct prime factors (A360458) we have A360552, complement A100367.
%Y A360681 - For prime factors (A360459) we have A359913, complement A072978.
%Y A360681 - For prime multiplicities (A360460) we have A360553, complement A360554.
%Y A360681 - For 0-prepended differences (A360555) we have A360556, complement A360557.
%Y A360681 Cf. A000975, A026424, A316413, A340610, A359904, A360006, A360248, A360558, A360687, A360688.
%K A360681 nonn
%O A360681 1,2
%A A360681 _Gus Wiseman_, Feb 19 2023