A360070
Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.
Original entry on oeis.org
1, 4, 8, 9, 12, 16, 18, 20, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 121, 125, 128, 144, 147, 150, 160, 162, 169, 175, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 245, 250, 252, 256, 272
Offset: 1
A partition of 20 with the same mean as its multiplicities is (5,4,3,2,1,1,1,1,1,1), so 20 is in the sequence.
A360680
Numbers for which the prime signature has the same mean as the first differences of 0-prepended prime indices.
Original entry on oeis.org
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
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.
For indices instead of 0-prepended differences:
A359903, counted by
A360068.
For median instead of mean we have
A360681.
A316413 = numbers whose prime indices have integer mean, complement
A348551.
A360614/
A360615 = mean of first differences of 0-prepended prime indices.
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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]]]&]
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