cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

A360553 Numbers > 1 whose unordered prime signature has integer median.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83
Offset: 1

Views

Author

Gus Wiseman, Feb 16 2023

Keywords

Comments

First differs from A067340 in having 60.
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).

Examples

			The unordered prime signature of 60 is {1,1,2}, with median 1, so 60 is in the sequence.
The unordered prime signature of 1260 is {1,1,2,2}, with median 3/2, so 1260 is not in the sequence.

Crossrefs

For mean instead of median we have A067340, complement A070011.
Positions of even terms in A360460.
The complement is A360554 (without 1).
These partitions are counted by A360687.
- 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.
A112798 lists prime indices, length A001222, sum A056239.
A124010 lists prime signature.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360454 = numbers whose prime indices and signature have the same median.
Cf. A000975, A026424, A078174, A078175, A316413, A326621, A360453.

Programs

  • Mathematica
    Select[Range[2,100],IntegerQ[Median[Last/@FactorInteger[#]]]&]

A360687 Number of integer partitions of n whose multiplicities have integer median.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 10, 16, 22, 34, 42, 65, 80, 115, 145, 195, 240, 324, 396, 519, 635, 814, 994, 1270, 1549, 1952, 2378, 2997, 3623, 4521, 5466, 6764, 8139, 10008, 12023, 14673, 17534, 21273, 25336, 30593, 36302, 43575, 51555, 61570, 72653, 86382, 101676
Offset: 1

Views

Author

Gus Wiseman, Feb 20 2023

Keywords

Comments

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

Examples

			The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (2211)    (3211)     (521)
                                     (3111)    (4111)     (2222)
                                     (111111)  (211111)   (3221)
                                               (1111111)  (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (221111)
                                                          (311111)
                                                          (11111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is counted under a(8).

Crossrefs

The case of an odd number of multiplicities is A090794.
For mean instead of median we have A360069, ranks A067340.
These partitions have ranks A360553.
The complement is counted by A360690, ranks A360554.
A058398 counts partitions by mean, see also A008284, A327482.
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.
Cf. A000975, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

A360552 Numbers > 1 whose distinct prime factors have integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 102, 103
Offset: 1

Views

Author

Gus Wiseman, Feb 16 2023

Keywords

Comments

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

Examples

			The prime factors of 900 are {2,2,3,3,5,5}, with distinct parts {2,3,5}, with median 3, so 900 is in the sequence.

Crossrefs

For mean instead of median we have A078174, complement of A176587.
The complement is A100367 (without 1).
Positions of even terms in A360458.
- 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.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A323171/A323172 = mean of distinct prime factors, indices A326619/A326620.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A056239, A078175, A175352, A316413, A326621, A360009.

Programs

  • Mathematica
    Select[Range[2,100],IntegerQ[Median[First/@FactorInteger[#]]]&]

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

Views

Author

Gus Wiseman, Feb 19 2023

Keywords

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).

Examples

			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.

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.

Programs

  • 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]]]&]

A360690 Number of integer partitions of n with non-integer median of multiplicities.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 5, 6, 8, 8, 14, 12, 21, 20, 31, 36, 57, 61, 94, 108, 157, 188, 261, 305, 409, 484, 632, 721, 942, 1083, 1376, 1585, 2004, 2302, 2860, 3304, 4103, 4742, 5849, 6745, 8281, 9599, 11706, 13605, 16481, 19176, 23078, 26838, 32145, 37387, 44465
Offset: 1

Views

Author

Gus Wiseman, Feb 22 2023

Keywords

Comments

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

Examples

			The a(1) = 0 through a(9) = 8 partitions:
  .  .  .  (211)  (221)  (411)    (322)    (332)      (441)
                  (311)  (21111)  (331)    (422)      (522)
                                  (511)    (611)      (711)
                                  (22111)  (22211)    (22221)
                                  (31111)  (41111)    (33111)
                                           (2111111)  (51111)
                                                      (2211111)
                                                      (3111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is not counted under a(8).

Crossrefs

These partitions have ranks A360554.
The complement is counted by A360687, ranks A360553.
A058398 counts partitions by mean, see also A008284, A327482.
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.
A360069 = partitions with integer mean of multiplicities, ranks A067340.
Cf. A000041, A000975, A090794, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

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

Views

Author

Gus Wiseman, Feb 19 2023

Keywords

Comments

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

Examples

			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.

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.

Programs

  • 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]]]&]
Previous Showing 11-16 of 16 results.

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