login
A360680
Numbers for which the prime signature has the same mean as the first differences of 0-prepended prime indices.
1
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.
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.
Sequence in context: A229882 A325986 A298759 * A369685 A127517 A137825
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 19 2023
STATUS
approved