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A360680 Numbers for which the prime signature has the same mean as the first differences of 0-prepended prime indices.

%I A360680 #6 Feb 20 2023 07:54:33
%S A360680 1,2,6,30,49,152,210,513,1444,1776,1952,2310,2375,2664,2760,2960,3249,
%T A360680 3864,3996,4140,4144,5796,5994,6072,6210,6440,6512,6517,6900,7176,
%U A360680 7400,7696,8694,9025,9108,9384,10064,10120,10350,10488,10764,11248,11960,12167
%N A360680 Numbers for which the prime signature has the same mean as the first differences of 0-prepended prime indices.
%C A360680 A number's (unordered) prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
%e A360680 The terms together with their prime indices begin:
%e A360680       1: {}
%e A360680       2: {1}
%e A360680       6: {1,2}
%e A360680      30: {1,2,3}
%e A360680      49: {4,4}
%e A360680     152: {1,1,1,8}
%e A360680     210: {1,2,3,4}
%e A360680     513: {2,2,2,8}
%e A360680    1444: {1,1,8,8}
%e A360680    1776: {1,1,1,1,2,12}
%e A360680    1952: {1,1,1,1,1,18}
%e A360680    2310: {1,2,3,4,5}
%e A360680    2375: {3,3,3,8}
%e A360680    2664: {1,1,1,2,2,12}
%e A360680    2760: {1,1,1,2,3,9}
%e A360680    2960: {1,1,1,1,3,12}
%e A360680 For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with mean 3/2. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with mean also 3/2. So 2760 is in the sequence.
%t A360680 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A360680 Select[Range[1000],Mean[Length/@Split[prix[#]]] == Mean[Differences[Prepend[prix[#],0]]]&]
%Y A360680 For indices instead of 0-prepended differences: A359903, counted by A360068.
%Y A360680 For median instead of mean we have A360681.
%Y A360680 A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
%Y A360680 A124010 gives prime signature, mean A088529/A088530.
%Y A360680 A316413 = numbers whose prime indices have integer mean, complement A348551.
%Y A360680 A326619/A326620 gives mean of distinct prime indices.
%Y A360680 A360614/A360615 = mean of first differences of 0-prepended prime indices.
%Y A360680 Cf. A340610, A359904, A359905, A360008, A360460, A360555, A360556.
%K A360680 nonn
%O A360680 1,2
%A A360680 _Gus Wiseman_, Feb 19 2023