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.

Showing 1-5 of 5 results.

A363945 Triangle read by rows where T(n,k) is the number of integer partitions of n with low mean k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 4, 2, 0, 0, 1, 0, 4, 3, 3, 0, 0, 1, 0, 7, 4, 3, 0, 0, 0, 1, 0, 7, 10, 0, 4, 0, 0, 0, 1, 0, 12, 6, 7, 4, 0, 0, 0, 0, 1, 0, 12, 16, 8, 0, 5, 0, 0, 0, 0, 1, 0, 19, 21, 10, 0, 5, 0, 0, 0, 0
Offset: 0

Views

Author

Gus Wiseman, Jun 30 2023

Keywords

Comments

Extending the terminology of A124943, the "low mean" of a multiset is its mean rounded down.

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  2  0  1
  0  2  2  0  1
  0  4  2  0  0  1
  0  4  3  3  0  0  1
  0  7  4  3  0  0  0  1
  0  7 10  0  4  0  0  0  1
  0 12  6  7  4  0  0  0  0  1
  0 12 16  8  0  5  0  0  0  0  1
  0 19 21 10  0  5  0  0  0  0  0  1
  0 19 24 15 12  0  6  0  0  0  0  0  1
  0 30 32 18 14  0  6  0  0  0  0  0  0  1
  0 30 58 23 16  0  0  7  0  0  0  0  0  0  1
  0 45 47 57  0 19  0  7  0  0  0  0  0  0  0  1
Row k = 8 counts the following partitions:
  .  (41111)     (611)   .  (71)  .  .  .  (8)
     (32111)     (521)      (62)
     (311111)    (5111)     (53)
     (22211)     (431)      (44)
     (221111)    (422)
     (2111111)   (4211)
     (11111111)  (332)
                 (3311)
                 (3221)
                 (2222)

Crossrefs

Row sums are A000041.
Column k = 1 is A025065, ranks A363949.
For median instead of mean we have triangle A124943, high A124944.
Column k = 2 is A363745.
For median instead of mean we have rank statistic A363941, high A363942.
The rank statistic for this triangle is A363943.
The high version is A363946, rank statistic A363944.
For mode instead of mean we have A363952, rank statistic A363486.
For high mode instead of mean we have A363953, rank statistic A363487.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A002865, A026905, A237984, A327472, A327482, A344296, A362612, A363723, A363724, A363731, A363951.

Programs

  • Mathematica
    meandown[y_]:=If[Length[y]==0,0,Floor[Mean[y]]];
Table[Length[Select[IntegerPartitions[n],meandown[#]==k&]],{n,0,15},{k,0,n}]

A363950 Numbers whose prime indices have rounded-up mean 2.

Original entry on oeis.org

3, 6, 9, 10, 12, 18, 20, 24, 27, 28, 30, 36, 40, 48, 54, 56, 60, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 144, 160, 162, 168, 176, 180, 192, 200, 208, 216, 224, 240, 243, 252, 264, 270, 280, 288, 300, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448
Offset: 1

Views

Author

Gus Wiseman, Jul 05 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:
     3: {2}
     6: {1,2}
     9: {2,2}
    10: {1,3}
    12: {1,1,2}
    18: {1,2,2}
    20: {1,1,3}
    24: {1,1,1,2}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    36: {1,1,2,2}
    40: {1,1,1,3}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    56: {1,1,1,4}
    60: {1,1,2,3}
    72: {1,1,1,2,2}
    80: {1,1,1,1,3}
    81: {2,2,2,2}

Crossrefs

For mean 1 we have A000079 except 1.
Partitions of this type are counted by A026905 redoubled.
Equals the complement of A000079 in A344296.
Positions of 2's in A363944 (counted by column 2 of A363946).
For rounded mean 1 we have A363948, counted by A363947.
For rounded-down mean 1 we have A363949, counted by A025065.
The rounded-down or low version is A363954, counted by A363745.
A316413 ranks partitions with integer mean, counted by A067538.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
Cf. A327473, A327476, A359889, A360005, A360013, A360015, A363727, A363943.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Ceiling[Mean[prix[#]]]==2&]

A363947 Number of integer partitions of n with mean < 3/2.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 4, 4, 4, 7, 7, 7, 12, 12, 12, 19, 19, 19, 30, 30, 30, 45, 45, 45, 67, 67, 67, 97, 97, 97, 139, 139, 139, 195, 195, 195, 272, 272, 272, 373, 373, 373, 508, 508, 508, 684, 684, 684, 915, 915, 915, 1212, 1212, 1212, 1597, 1597, 1597, 2087
Offset: 0

Views

Author

Gus Wiseman, Jul 02 2023

Keywords

Examples

			The partition y = (2,2,1) has mean 5/3, which is not less than 3/2, so y is not counted under 5.
The a(1) = 1 through a(8) = 4 partitions:
  (1)  (11)  (111)  (211)   (2111)   (21111)   (22111)    (221111)
                    (1111)  (11111)  (111111)  (31111)    (311111)
                                               (211111)   (2111111)
                                               (1111111)  (11111111)

Crossrefs

The high version is A000012 (all ones).
This is A000070 with each term repeated three times (see A025065 for two).
These partitions have ranks A363948.
The complement is counted by A364059.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A327482 counts partitions by integer mean.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A000041, A002865, A026905, A027336, A237984, A241131, A327472, A363724, A363745, A363943, A363949.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Round[Mean[#]]==1&]],{n,0,15}]

A363954 Numbers whose prime indices have low mean 2.

Original entry on oeis.org

3, 9, 10, 14, 15, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 70, 75, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 189, 196, 198, 204, 208, 210, 220, 225, 234, 243, 250, 252, 260, 264, 270, 272, 280, 294, 297, 300, 304, 308, 312, 315, 330, 350
Offset: 1

Views

Author

Gus Wiseman, Jul 05 2023

Keywords

Comments

Extending the terminology of A124944, the "low mean" of a multiset is obtained by taking the mean and rounding down.

Examples

			The terms together with their prime indices begin:
     3: {2}
     9: {2,2}
    10: {1,3}
    14: {1,4}
    15: {2,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    42: {1,2,4}
    44: {1,1,5}
    45: {2,2,3}
    50: {1,3,3}
    52: {1,1,6}
    63: {2,2,4}
    66: {1,2,5}
    70: {1,3,4}
    75: {2,3,3}
    81: {2,2,2,2}
    84: {1,1,2,4}
    88: {1,1,1,5}
    90: {1,2,2,3}
   100: {1,1,3,3}

Crossrefs

Partitions of this type are counted by A363745.
Positions of 2's in A363943 (high A363944), triangle A363945 (high A363946).
For mean 1 we have A363949.
The high version is A363950, counted by A026905.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A326567/A326568 gives mean of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
A363948 lists numbers whose prime indices have mean 1, counted by A363947.
Cf. A327473, A327476, A359889, A360013, A360015, A363488, A363951.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Floor[Mean[prix[#]]]==2&]

A364157 Numbers whose rounded-down (floor) mean of prime factors (with multiplicity) is 2.

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 40, 48, 54, 64, 72, 80, 96, 108, 120, 128, 144, 160, 162, 192, 216, 224, 240, 256, 288, 320, 324, 360, 384, 432, 448, 480, 486, 512, 576, 640, 648, 672, 720, 768, 800, 864, 896, 960, 972, 1024, 1080, 1152, 1280, 1296, 1344
Offset: 1

Views

Author

Gus Wiseman, Jul 18 2023

Keywords

Examples

			The terms together with their prime factors begin:
   2 = 2
   4 = 2*2
   6 = 2*3
   8 = 2*2*2
  12 = 2*2*3
  16 = 2*2*2*2
  18 = 2*3*3
  24 = 2*2*2*3
  32 = 2*2*2*2*2
  36 = 2*2*3*3
  40 = 2*2*2*5
  48 = 2*2*2*2*3
  54 = 2*3*3*3
  64 = 2*2*2*2*2*2
  72 = 2*2*2*3*3
  80 = 2*2*2*2*5
  96 = 2*2*2*2*2*3

Crossrefs

Without multiplicity we appear to have A007694.
Prime factors are listed by A027746, indices A112798.
Positions of 2's in A126594, positions of first appearances A364037.
For prime indices and ceiling we have A363950, counted by A026905.
For prime indices we have A363954 (or A363949), counted by A363745.
A078175 lists numbers with integer mean of prime factors.
A123528/A123529 gives mean of prime factors, indices A326567/A326568.
A316413 ranks partitions with integer mean, counted by A067538.
A363895 gives floor of mean of distinct prime factors.
A363943 gives floor of mean of prime indices, ceiling A363944.
Cf. A001222, A056239, A327473, A327476, A360013, A360015, A363488, A363945.

Programs

  • Mathematica
    prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],Floor[Mean[prifacs[#]]]==2&]
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.