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 1-4 of 4 results.

A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 1, 2, 4, 0, 10, 0, 8, 16, 10, 0, 31, 0, 44, 44, 20, 0, 92, 50, 28, 98, 154, 0, 266, 0, 154, 194, 48, 434, 712, 0, 60, 348, 910, 0, 1198, 0, 1120, 2138, 88, 0, 2428, 1300, 1680, 912, 2506, 0, 4808, 4800, 5968, 1372, 140, 0, 14820, 0, 160
Offset: 1

Views

Author

Gus Wiseman, Mar 29 2023

Keywords

Comments

Also partitions satisfying (maximum) = 2*(mean).
These are partitions whose diagram has the same size as its complement (see example).

Examples

			The a(6) = 2 through a(12) = 10 partitions:
  (411)   .  (4211)  (621)     (5221)   .  (822)
  (3111)             (321111)  (5311)      (831)
                               (42211)     (6222)
                               (43111)     (6321)
                                           (6411)
                                           (422211)
                                           (432111)
                                           (441111)
                                           (32211111)
                                           (33111111)
The partition y = (6,4,1,1) has diagram:
  o o o o o o
  o o o o . .
  o . . . . .
  o . . . . .
Since the partition and its complement (shown in dots) have the same size, y is counted under a(12).

Crossrefs

For minimum instead of mean we have A118096.
For length instead of mean we have A237753.
For median instead of mean we have A361849, ranks A361856.
This is the equal case of A361851, unequal case A361852.
The strict case is A361854.
These partitions have ranks A361855.
This is the equal case of A361906, unequal case A361907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
A268192 counts partitions by complement size, ranks A326844.
Cf. A111907, A116608, A188814, A237755, A237824, A237984, A240219, A326849, A327482, A349156, A359894.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#==2n&]],{n,30}]

A361855 Numbers > 1 whose prime indices satisfy (maximum) * (length) = 2*(sum).

Original entry on oeis.org

28, 40, 78, 84, 171, 190, 198, 220, 240, 252, 280, 351, 364, 390, 406, 435, 714, 748, 756, 765, 777, 784, 814, 840, 850, 925, 988, 1118, 1197, 1254, 1330, 1352, 1419, 1425, 1440, 1505, 1564, 1600, 1638, 1716, 1755, 1794, 1802, 1820, 1950, 2067, 2204, 2254
Offset: 1

Views

Author

Gus Wiseman, Mar 29 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.
Also positive integers whose prime indices satisfy (maximum) = 2*(mean).
Also Heinz numbers of partitions of the same size as their complement (see example).

Examples

			The terms together with their prime indices begin:
   28: {1,1,4}
   40: {1,1,1,3}
   78: {1,2,6}
   84: {1,1,2,4}
  171: {2,2,8}
  190: {1,3,8}
  198: {1,2,2,5}
  220: {1,1,3,5}
  240: {1,1,1,1,2,3}
  252: {1,1,2,2,4}
  280: {1,1,1,3,4}
The prime indices of 84 are {1,1,2,4}, with maximum 4, length 4, and sum 8, and 4*4 = 2*8, so 84 is in the sequence.
The prime indices of 120 are {1,1,1,2,3}, with maximum 3, length 5, and sum 8, and 3*5 != 2*8, so 120 is not in the sequence.
The prime indices of 252 are {1,1,2,2,4}, with maximum 4, length 5, and sum 10, and 4*5 = 2*10, so 252 is in the sequence.
The partition (5,2,2,1) with Heinz number 198 has diagram:
  o o o o o
  o o . . .
  o o . . .
  o . . . .
Since the partition and its complement (shown in dots) both have size 10, 198 is in the sequence.

Crossrefs

These partitions are counted by A361853, strict A361854.
For median instead of mean we have A361856, counted by A361849.
For minimum instead of mean we have A361908, counted by A118096.
For length instead of mean we have A361909, counted by A237753.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A061395 gives greatest prime index.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
Cf. A067801, A237824, A316413, A324521, A326844, A361205, A361851, A361906.

Programs

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

Formula

A061395(a(n)) * A001222(a(n)) = 2*A056239(a(n)).

A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 15, 14, 32, 34, 65, 55, 150, 100, 225, 237, 425, 296, 824, 489, 1267, 1133, 1809, 1254, 4018, 2142, 4499, 4550, 7939, 4564, 14571, 6841, 18285, 16047, 23408, 17495, 52545, 21636, 49943, 51182, 92516, 44582, 144872, 63260, 175318, 169232, 205353
Offset: 0

Views

Author

Gus Wiseman, May 23 2023

Keywords

Comments

Equivalently, n = (length)*(minimum).

Examples

			The a(2) = 1 through a(7) = 14 partitions:
  (31)  (321)  (62)    (32221)  (93)      (3222221)
        (411)  (3221)  (33211)  (552)     (3322211)
               (3311)  (42211)  (642)     (3332111)
               (4211)  (43111)  (732)     (4222211)
               (5111)  (52111)  (822)     (4322111)
                       (61111)  (322221)  (4331111)
                                (332211)  (4421111)
                                (333111)  (5222111)
                                (422211)  (5321111)
                                (432111)  (5411111)
                                (441111)  (6221111)
                                (522111)  (6311111)
                                (531111)  (7211111)
                                (621111)  (8111111)
                                (711111)

Crossrefs

Removing the factor 2 gives A099777.
Taking maximum instead of mean and including odd indices gives A118096.
For length instead of mean and including odd indices we have A237757.
For (maximum) = 2*(mean) see A361851, A361852, A361853, A361854, A361855.
For median instead of mean we have A361861.
These partitions have ranks A363133.
For maximum instead of minimum we have A363218.
For median instead of minimum we have A363224.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
A268192 counts partitions by complement size, ranks A326844.
Cf. A053263, A111907, A237753, A237755, A237824, A327482, A349156, A361906, A361907, A363134.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],2*Min@@#==Mean[#]&]],{n,0,15}]
  • Python
    from sympy.utilities.iterables import partitions
def A363132(n): return sum(1 for s,p in partitions(n<<1,m=n,size=True) if n==s*min(p,default=0)) if n else 0 # Chai Wah Wu, Sep 21 2023

Extensions

a(31)-a(46) from Chai Wah Wu, Sep 21 2023

A363221 Number of strict integer partitions of n such that (length) * (maximum) <= 2n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 19, 23, 26, 29, 37, 39, 49, 55, 62, 71, 84, 93, 108, 118, 141, 149, 188, 193, 217, 257, 279, 318, 369, 376, 441, 495, 572, 587, 692, 760, 811, 960, 1046, 1065, 1307, 1387, 1550, 1703, 1796, 2041, 2295, 2456, 2753, 3014
Offset: 1

Views

Author

Gus Wiseman, May 23 2023

Keywords

Comments

Also strict partitions such that (maximum) <= 2*(mean).
These are strict partitions whose complement (see A361851) has size <= n.

Examples

			The partition y = (4,3,1) has length 3 and maximum 4, and 3*4 <= 2*8, so y is counted under a(8). The complement of y has size 4, which is less than or equal to n = 8.

Crossrefs

The equal case for median is A361850, non-strict A361849 (ranks A361856).
The non-strict version is A361851, A361848 for median.
The equal case is A361854, non-strict A361853 (ranks A361855).
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
Cf. A111907, A237984, A240219, A241061, A241086, A324521, A324562, A349156, A360068, A360241, A361394, A361852, A361906.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Max@@#<=2*Mean[#]&]],{n,30}]
Showing 1-4 of 4 results.

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