A360248 -id:A360248 - cp's OEIS frontend

A360456 Number of integer partitions of n for which the parts have the same median as the multiplicities.

1, 1, 0, 0, 1, 0, 0, 1, 2, 5, 7, 10, 14, 21, 28, 36, 51, 64, 84, 106, 132, 165, 202, 252, 311, 391, 473, 579, 713, 868, 1069, 1303, 1617, 1954, 2404, 2908, 3556, 4282, 5200, 6207, 7505, 8934, 10700, 12717, 15165, 17863, 21222, 24976, 29443, 34523, 40582, 47415

Offset: 0

Gus Wiseman, Feb 10 2023

nonn

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

			The a(1) = 1 through a(11) = 10 partitions:
  1   .  .  22   .  .  2221   3311    333      4222      5222
                              32111   3222     33211     33221
                                      32211    42211     52211
                                      42111    43111     53111
                                      321111   52111     62111
                                               421111    322211
                                               3211111   431111
                                                         521111
                                                         4211111
                                                         32111111

For mean instead of median: A360068, ranks A359903.
For distinct parts instead of multiplicities: A360245, ranks A360249.
These partitions have ranks A360454.
For distinct parts instead of parts: A360455, ranks A360453.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A114638, A240219, A359894, A359895, A360244, A360248.

Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[#]&]],{n,0,30}]

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

Original entry on oeis.org

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

Gus Wiseman, Feb 19 2023

nonn

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).

			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.

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.

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]]]&]

A376250 Numbers with a unique largest prime exponent (A356862) that are not prime powers (A246655).

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 198, 200

Offset: 1

Amiram Eldar, Sep 17 2024

First differs from A059404 at n = 55: A059404(55) = 180 = 2^2 * 3^2 * 5 is not a term of this sequence.
First differs from A360248 at n = 23: a(23) = 90 = 2 * 3^2 * 5 is not a term of A360248.
First differs from A332785 at n = 17: a(17) = 72 = 2^3 * 3^2 is not a term of A332785.
Numbers whose unordered prime signature (i.e., sorted, see A118914) ends with two different integers: {..., k, m} for some 1 <= k < m.
All the factorial numbers above 6 are terms.
The asymptotic density of this sequence is Sum_{k >= 1, p prime} (d(k+1, p) - d(k, p))/((p-1)*p^k) = 0.3660366524547281232052..., where d(k, p) = 0 for k = 1, and (1-1/p)/((1-1/p^k)*zeta(k)) for k > 1, is the density of terms that have in their prime factorization a prime p with the largest exponent that is > k.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Equals A356862 \ A246655.

Cf. A000142, A059404, A118914, A332785, A360248.

Select[Range[2, 200], Length[e = FactorInteger[#][[;; , 2]]] > 1 &&  Count[e, Max[e]] == 1 &]

is(k) = if (k == 1, 0, my(e = vecsort(factor(k)[,2])); #e > 1 && e[#e] > e[#e-1]);

A363220 Number of integer partitions of n whose conjugate has the same median.

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 3, 8, 8, 12, 12, 15, 21, 27, 36, 49, 65, 85, 112, 149, 176, 214, 257, 311, 378, 470, 572, 710, 877, 1080, 1322, 1637, 1983, 2416, 2899, 3465, 4107, 4891, 5763, 6820, 8071, 9542, 11289, 13381, 15808, 18710, 22122, 26105, 30737, 36156, 42377

Offset: 1

Gus Wiseman, May 29 2023

nonn

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

			The partition y = (4,3,1,1) has median 2, and its conjugate (4,2,2,1) also has median 2, so y is counted under a(9).
The a(1) = 1 through a(9) = 8 partitions:
  (1)  .  (21)  (22)  (311)  (321)   (511)    (332)     (333)
                             (411)   (4111)   (422)     (711)
                             (3111)  (31111)  (611)     (4221)
                                              (3311)    (4311)
                                              (4211)    (6111)
                                              (5111)    (51111)
                                              (41111)   (411111)
                                              (311111)  (3111111)

For mean instead of median we have A047993.
For product instead of median we have A325039, ranks A325040.
For union instead of conjugate we have A360245, complement A360244.
Median of conjugate by rank is A363219.
These partitions are ranked by A363261.
A000700 counts self-conjugate partitions, ranks A088902.
A046682 and A352487-A352490 pertain to excedance set.
A122111 represents partition conjugation.
A325347 counts partitions with integer median.
A330644 counts non-self-conjugate partitions (twice A000701), ranks A352486.
A352491 gives n minus Heinz number of conjugate.
Cf. A000975, A067538, A114638, A360068, A360242, A360248, A362617, A362618, A362621, A363223, A363260.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Median[#]==Median[conj[#]]&]],{n,30}]

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A360456 Number of integer partitions of n for which the parts have the same median as the multiplicities.

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

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A376250 Numbers with a unique largest prime exponent (A356862) that are not prime powers (A246655).

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A363220 Number of integer partitions of n whose conjugate has the same median.

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