A360680 - OEIS

%I #6 Feb 20 2023 07:54:33

%S 1,2,6,30,49,152,210,513,1444,1776,1952,2310,2375,2664,2760,2960,3249,

%T 3864,3996,4140,4144,5796,5994,6072,6210,6440,6512,6517,6900,7176,

%U 7400,7696,8694,9025,9108,9384,10064,10120,10350,10488,10764,11248,11960,12167

%N Numbers for which the prime signature has the same mean 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.

%e The terms together with their prime indices begin:

%e       1: {}

%e       2: {1}

%e       6: {1,2}

%e      30: {1,2,3}

%e      49: {4,4}

%e     152: {1,1,1,8}

%e     210: {1,2,3,4}

%e     513: {2,2,2,8}

%e    1444: {1,1,8,8}

%e    1776: {1,1,1,1,2,12}

%e    1952: {1,1,1,1,1,18}

%e    2310: {1,2,3,4,5}

%e    2375: {3,3,3,8}

%e    2664: {1,1,1,2,2,12}

%e    2760: {1,1,1,2,3,9}

%e    2960: {1,1,1,1,3,12}

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

%t Select[Range[1000],Mean[Length/@Split[prix[#]]] == Mean[Differences[Prepend[prix[#],0]]]&]

%Y For indices instead of 0-prepended differences: A359903, counted by A360068.

%Y For median instead of mean we have A360681.

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

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

%Y A316413 = numbers whose prime indices have integer mean, complement A348551.

%Y A326619/A326620 gives mean of distinct prime indices.

%Y A360614/A360615 = mean of first differences of 0-prepended prime indices.

%Y Cf. A340610, A359904, A359905, A360008, A360460, A360555, A360556.

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 19 2023