A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

1, 2, 6, 30, 42, 49, 60, 66, 70, 78, 84, 90, 102, 105, 114, 120, 126, 132, 138, 140, 150, 154, 156, 168, 174, 186, 198, 204, 210, 222, 228, 234, 246, 258, 264, 270, 276, 280, 282, 286, 294, 306, 308, 312, 315, 318, 330, 342, 348, 350, 354, 366, 372, 378, 385

Offset

1,2

Comments

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

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Links

Table of n, a(n) for n=1..55.

Example

The terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
   30: {1,2,3}
   42: {1,2,4}
   49: {4,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with median 1. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with median 1/2. So 2760 is not in the sequence.

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Median[Length/@Split[prix[#]]] == Median[Differences[Prepend[prix[#], 0]]]&]

Crossrefs

For distinct prime indices instead of 0-prepended differences: A360453.

For mean instead of median we have A360680.

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

A124010 gives prime signature, sorted A118914, mean A088529/A088530.

A325347 = partitions w/ integer median, strict A359907, complement A307683.

A359893 and A359901 count partitions by median, odd-length A359902.

Multisets with integer median:

- For divisors (A063655) we have A139711, complement A139710.

- For prime indices (A360005) we have A359908, complement A359912.

- For distinct prime indices (A360457) we have A360550, complement A360551.

- For distinct prime factors (A360458) we have A360552, complement A100367.

- For prime factors (A360459) we have A359913, complement A072978.

- For prime multiplicities (A360460) we have A360553, complement A360554.

- For 0-prepended differences (A360555) we have A360556, complement A360557.

Cf. A000975, A026424, A316413, A340610, A359904, A360006, A360248, A360558, A360687, A360688.

Sequence in context: A346106 A330326 A006954 * A286652 A265501 A090801

Adjacent sequences: A360678 A360679 A360680 * A360682 A360683 A360684

Keyword

nonn

Author

Gus Wiseman, Feb 19 2023

Status

approved