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 21-27 of 27 results.

A360251 Number of integer partitions of n where the parts have lesser mean than the distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 7, 9, 16, 22, 34, 44, 69, 88, 118, 163, 221, 280, 376, 473, 619, 800, 1016, 1257, 1621, 2038, 2522, 3117, 3921, 4767, 5964, 7273, 8886, 10838, 13141, 15907, 19468, 23424, 28093, 33656, 40672, 48273, 58171, 68944, 81888, 97596, 115643
Offset: 0

Views

Author

Gus Wiseman, Feb 06 2023

Keywords

Examples

			The a(4) = 1 through a(9) = 16 partitions:
  (211)  (311)   (411)    (322)     (422)      (522)
         (2111)  (3111)   (511)     (611)      (711)
                 (21111)  (3211)    (4211)     (3222)
                          (4111)    (5111)     (4221)
                          (22111)   (32111)    (4311)
                          (31111)   (41111)    (5211)
                          (211111)  (221111)   (6111)
                                    (311111)   (32211)
                                    (2111111)  (33111)
                                               (42111)
                                               (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
For example, the partition y = (4,2,2,1) has mean 9/4 and distinct parts {1,2,4} with mean 7/3, so y is counted under a(9).

Crossrefs

For unequal instead of less we have A360242, ranks A360246.
For equal instead of less we have A360243, ranks A360247.
For greater instead of less we have A360250, ranks A360252.
These partitions have ranks A360253.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A058398 counts partitions by mean, also A327482.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A240219 counts partitions with mean equal to median, ranks A359889.
A359894 counts partitions with mean different from median, ranks A359890.
A360071 counts partitions by number of parts and number of distinct parts.
Cf. A000975, A316313, A326567/A326568, A326619/A326620, A326621, A360068, A360241, A360244, A360245.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Mean[#]

Formula

a(n) + A360250(n) = A360242(n).
a(n) + A360250(n) + A360243(n) = A000041(n).

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[#]]]&]

A360252 Numbers for which the prime indices have greater mean than the distinct prime indices.

Original entry on oeis.org

18, 50, 54, 75, 98, 108, 147, 150, 162, 242, 245, 250, 294, 324, 338, 350, 363, 375, 450, 486, 490, 500, 507, 578, 588, 605, 648, 686, 722, 726, 735, 750, 845, 847, 867, 882, 972, 1014, 1029, 1050, 1058, 1078, 1083, 1125, 1183, 1210, 1250, 1274, 1350, 1372
Offset: 1

Views

Author

Gus Wiseman, Feb 09 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The terms together with their prime indices begin:
    18: {1,2,2}
    50: {1,3,3}
    54: {1,2,2,2}
    75: {2,3,3}
    98: {1,4,4}
   108: {1,1,2,2,2}
   147: {2,4,4}
   150: {1,2,3,3}
   162: {1,2,2,2,2}
   242: {1,5,5}
   245: {3,4,4}
   250: {1,3,3,3}
   294: {1,2,4,4}
   324: {1,1,2,2,2,2}
For example, the prime indices of 350 are {1,3,3,4} with mean 11/4, and the distinct prime indices are {1,3,4} with mean 8/3, so 350 is in the sequence.

Crossrefs

For unequal instead of greater we have A360246, counted by A360242.
For equal instead of greater we have A360247, counted by A360243.
These partitions are counted by A360250.
For less instead of greater we have A360253, counted by A360251.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose indices have integer mean, distinct A326621.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
Cf. A000975, A051293, A058398, A067340, A067538, A324570, A327482, A359903, A360005, A360241, A360248.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Mean[prix[#]]>Mean[Union[prix[#]]]&]

A360253 Numbers for which the prime indices have lesser mean than the distinct prime indices.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212, 220
Offset: 1

Views

Author

Gus Wiseman, Feb 09 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The terms together with their prime indices begin:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   68: {1,1,7}
   72: {1,1,1,2,2}
For example, the prime indices of 350 are {1,3,3,4} with mean 11/4, and the distinct prime indices are {1,3,4} with mean 8/3, so 350 is not in the sequence.

Crossrefs

These partitions are counted by A360251.
For unequal instead of less we have A360246, counted by A360242.
For equal instead of less we have A360247, counted by A360243.
For greater instead of less we have A360252, counted by A360250.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose indices have integer mean, distinct A326621.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
Cf. A000975, A051293, A058398, A067340, A067538, A324570, A327482, A359903, A360005, A360241, A360248.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Mean[prix[#]]

A360454 Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.

Original entry on oeis.org

1, 2, 9, 54, 100, 120, 125, 135, 168, 180, 189, 240, 252, 264, 280, 297, 300, 312, 336, 351, 396, 408, 440, 450, 456, 459, 468, 480, 513, 520, 528, 540, 552, 560, 588, 612, 616, 621, 624, 672, 680, 684, 696, 728, 744, 756, 760, 783, 816, 828, 837, 880, 882
Offset: 1

Views

Author

Gus Wiseman, Feb 10 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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}
    9: {2,2}
   54: {1,2,2,2}
  100: {1,1,3,3}
  120: {1,1,1,2,3}
  125: {3,3,3}
  135: {2,2,2,3}
  168: {1,1,1,2,4}
  180: {1,1,2,2,3}
  189: {2,2,2,4}
  240: {1,1,1,1,2,3}
For example, the prime indices of 336 are {1,1,1,1,2,4} with median 1 and multiplicities {1,1,4} with median 1, so 336 is in the sequence.

Crossrefs

For mean instead of median we have A359903, counted by A360068.
For distinct indices instead of indices we have A360453, counted by A360455.
For distinct indices instead of multiplicities: A360249, counted by A360245.
These partitions are counted by A360456.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A240219 counts partitions with mean equal to median, ranked by A359889.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median.
A359894 counts partitions with mean different from median, ranks A359890.
A360005 gives median of prime indices (times two).
Cf. A000975, A109297, A109298, A114638, A316413, A324570, A360244, A360248.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Median[prix[#]]==Median[Length/@Split[prix[#]]]&]

A360455 Number of integer partitions of n for which the distinct parts have the same median as the multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 1, 0, 2, 2, 5, 8, 10, 14, 20, 19, 26, 31, 35, 41, 55, 65, 85, 102, 118, 151, 181, 201, 236, 281, 313, 365, 424, 495, 593, 688, 825, 978, 1181, 1374, 1650, 1948, 2323, 2682, 3175, 3680, 4314, 4930, 5718, 6546, 7532, 8557, 9777, 11067, 12622
Offset: 0

Views

Author

Gus Wiseman, Feb 10 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(11) = 8 partitions:
  1   .  .  22    221   3111   .  3311    333     3331     32222
            211                   41111   32211   33211    33221
                                                  42211    44111
                                                  322111   52211
                                                  511111   322211
                                                           332111
                                                           422111
                                                           3221111

Crossrefs

For mean instead of median: A114638, ranks A324570.
For parts instead of multiplicities: A360245, ranks A360249.
These partitions have ranks A360453.
For parts instead of distinct parts: A360456, ranks A360454.
A000041 counts integer partitions, strict A000009.
A116608 counts partitions by number of distinct parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A240219, A326619/A326620, A359894, A359895, A360068, A360243, A360244, A360248.

Programs

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

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 21-27 of 27 results.

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