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

1, 2, 6, 30, 49, 152, 210, 513, 1444, 1776, 1952, 2310, 2375, 2664, 2760, 2960, 3249, 3864, 3996, 4140, 4144, 5796, 5994, 6072, 6210, 6440, 6512, 6517, 6900, 7176, 7400, 7696, 8694, 9025, 9108, 9384, 10064, 10120, 10350, 10488, 10764, 11248, 11960, 12167

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..44.

Example

The terms together with their prime indices begin:
      1: {}
      2: {1}
      6: {1,2}
     30: {1,2,3}
     49: {4,4}
    152: {1,1,1,8}
    210: {1,2,3,4}
    513: {2,2,2,8}
   1444: {1,1,8,8}
   1776: {1,1,1,1,2,12}
   1952: {1,1,1,1,1,18}
   2310: {1,2,3,4,5}
   2375: {3,3,3,8}
   2664: {1,1,1,2,2,12}
   2760: {1,1,1,2,3,9}
   2960: {1,1,1,1,3,12}
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.

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Mean[Length/@Split[prix[#]]] == Mean[Differences[Prepend[prix[#], 0]]]&]

Crossrefs

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

For median instead of mean we have A360681.

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

A124010 gives prime signature, mean A088529/A088530.

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

A326619/A326620 gives mean of distinct prime indices.

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

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

Sequence in context: A229882 A325986 A298759 * A369685 A127517 A137825

Adjacent sequences: A360677 A360678 A360679 * A360681 A360682 A360683

Keyword

nonn

Author

Gus Wiseman, Feb 19 2023

Status

approved