A118096 -id:A118096 - cp's OEIS frontend

A239500 Number of partitions p of n such that if h = (number of parts of p), then h is an (h,1)-separator of p; see Comments.

0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 18, 20, 22, 24, 27, 29, 32, 36, 39, 43, 48, 53, 58, 65, 70, 78, 85, 93, 101, 112, 120, 132, 143, 156, 168, 184, 198, 216, 233, 253, 273, 298, 320, 348, 376, 407, 439

Offset: 1

Clark Kimberling, Mar 24 2014

Suppose that p is a partition of n into 2 or more parts and that h is a part of p. Then p is (h,0)-separable if there is an ordering x, h, x, h, ..., h, x of the parts of p, where each x represents any part of p except h. Here, the number of h's on the ends of the ordering is 0. Similarly, p is (h,1)-separable if there is an ordering x, h, x, h, ..., x, h, where the number of h's on the ends is 1; next, p is (h,2)-separable if there is an ordering h, x, h, ..., x, h. Finally, p is h-separable if it is (h,i)-separable for i = 0,1,2.

			a(12) counts these partitions: 84, 4431, 4422.

Cf. A239497, A239498, A118096, A239501, A239482.

z = 35; t1 = Table[Count[IntegerPartitions[n],  p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}]  (* A239497 *)
t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}]  (* A239501 *)

A239501 Number of partitions p of n such that if h = max(p) - min(p), then h is an (h,1)-separator of p; see Comments.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 3, 3, 2, 2, 3, 5, 4, 8, 4, 5, 9, 6, 13, 10, 11, 15, 14, 17, 16, 20, 21, 26, 29, 30, 33, 36, 35, 41, 47, 47, 61, 61, 66, 71, 73, 85, 88, 98, 102, 114, 122, 131, 148, 154, 163, 182, 188, 205, 220, 231, 249, 271, 293, 306, 338, 359

Offset: 1

Clark Kimberling, Mar 24 2014

Suppose that p is a partition of n into 2 or more parts and that h is a part of p. Then p is (h,0)-separable if there is an ordering x, h, x, h, ..., h, x of the parts of p, where each x represents any part of p except h. Here, the number of h's on the ends of the ordering is 0. Similarly, p is (h,1)-separable if there is an ordering x, h, x, h, ..., x, h, where the number of h's on the ends is 1; next, p is (h,2)-separable if there is an ordering h, x, h, ..., x, h. Finally, p is h-separable if it is (h,i)-separable for i = 0,1,2.

			a(11) counts these partitions: 4313, 4232, 321212.

Cf. A239497, A239498, A239499, A239500, A239482.

z = 35; t1 = Table[Count[IntegerPartitions[n],  p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}]  (* A239497 *)
t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}]  (* A239501 *)

A361854 Number of strict integer partitions of n such that (length) * (maximum) = 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 2, 0, 5, 0, 6, 3, 5, 0, 11, 6, 8, 7, 10, 0, 36, 0, 14, 16, 16, 29, 43, 0, 21, 36, 69, 0, 97, 0, 35, 138, 33, 0, 150, 61, 137, 134, 74, 0, 231, 134, 265, 229, 56, 0, 650, 0, 65, 749, 267, 247, 533, 0, 405, 565

Offset: 1

Gus Wiseman, Mar 29 2023

nonn

Also strict partitions satisfying (maximum) = 2*(mean).

These are strict partitions where both the diagram and its complement (see example) have size n.

			The a(n) strict partitions for selected n (A..E = 10..14):
  n=9:  n=12:  n=14:  n=15:  n=16:  n=18:  n=20:  n=21:  n=22:
--------------------------------------------------------------
  621   831    7421   A32    8431   C42    A532   E43    B542
        6321          A41    8521   C51    A541   E52    B632
                                    9432   A631   E61    B641
                                    9531   A721          B731
                                    9621   85421         B821
                                           86321
The a(20) = 6 strict partitions are: (10,7,2,1), (10,6,3,1), (10,5,4,1), (10,5,3,2), (8,6,3,2,1), (8,5,4,2,1).
The strict partition y = (8,5,4,2,1) has diagram:
  o o o o o o o o
  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(20).

For minimum instead of mean we have A241035, non-strict A118096.
For length instead of mean we have A241087, non-strict A237753.
For median instead of mean we have A361850, non-strict A361849.
The non-strict version is A361853.
These partitions have ranks A361855 /\ A005117.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A008289 counts strict partitions by length.
A102627 counts strict partitions with integer mean, non-strict A067538.
A116608 counts partitions by number of distinct parts.
A268192 counts partitions by complement size, ranks A326844.
Cf. A111907, A237755, A240850, A326849 A359897, A360068, A360071, A360243, A361848, A361851, A361852, A361906.

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

A361861 Number of integer partitions of n where the median is twice the minimum.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 5, 5, 8, 11, 16, 20, 28, 38, 53, 67, 87, 111, 146, 183, 236, 297, 379, 471, 591, 729, 909, 1116, 1376, 1682, 2065, 2507, 3055, 3699, 4482, 5395, 6501, 7790, 9345, 11153, 13316, 15839, 18844, 22333, 26466, 31266, 36924, 43478, 51177

Offset: 1

Gus Wiseman, Apr 02 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(4) = 1 through a(11) = 11 partitions:
  (31)  (221)  (321)  (421)   (62)     (621)    (442)     (542)
                      (2221)  (521)    (4221)   (721)     (821)
                              (3221)   (4311)   (5221)    (6221)
                              (3311)   (22221)  (5311)    (6311)
                              (22211)  (32211)  (32221)   (33221)
                                                (33211)   (42221)
                                                (42211)   (43211)
                                                (222211)  (52211)
                                                          (222221)
                                                          (322211)
                                                          (2222111)
The partition (3,2,2,2,1,1) has median 2 and minimum 1, so is counted under a(11).
The partition (5,4,2) has median 4 and minimum 2, so is counted under a(11).

For maximum instead of median we have A118096.
For length instead of median we have A237757, without the coefficient A006141.
With minimum instead of twice minimum we have A361860.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A027193, A039900, A053263, A067659, A111907, A116608, A237753, A237755, A237824, A361848, A361853.

Table[Length[Select[IntegerPartitions[n],2*Min@@#==Median[#]&]],{n,30}]

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

Gus Wiseman, May 23 2023

nonn

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

			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)

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.

Table[Length[Select[IntegerPartitions[2n],2*Min@@#==Mean[#]&]],{n,0,15}]

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

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

A363134 Positive integers whose multiset of prime indices satisfies: (length) = 2*(minimum).

Original entry on oeis.org

4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 81, 82, 86, 94, 106, 118, 122, 134, 135, 142, 146, 158, 166, 178, 189, 194, 202, 206, 214, 218, 225, 226, 254, 262, 274, 278, 297, 298, 302, 314, 315, 326, 334, 346, 351, 358, 362, 375, 382, 386, 394, 398, 422, 441

Offset: 1

Gus Wiseman, Jun 05 2023

nonn

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 terms together with their prime indices begin:
     4: {1,1}         94: {1,15}       214: {1,28}
     6: {1,2}        106: {1,16}       218: {1,29}
    10: {1,3}        118: {1,17}       225: {2,2,3,3}
    14: {1,4}        122: {1,18}       226: {1,30}
    22: {1,5}        134: {1,19}       254: {1,31}
    26: {1,6}        135: {2,2,2,3}    262: {1,32}
    34: {1,7}        142: {1,20}       274: {1,33}
    38: {1,8}        146: {1,21}       278: {1,34}
    46: {1,9}        158: {1,22}       297: {2,2,2,5}
    58: {1,10}       166: {1,23}       298: {1,35}
    62: {1,11}       178: {1,24}       302: {1,36}
    74: {1,12}       189: {2,2,2,4}    314: {1,37}
    81: {2,2,2,2}    194: {1,25}       315: {2,2,3,4}
    82: {1,13}       202: {1,26}       326: {1,38}
    86: {1,14}       206: {1,27}       334: {1,39}

Partitions of this type are counted by A237757.
Removing the factor 2 gives A324522.
For maximum instead of length we have A361908, counted by A118096.
For mean instead of length we have A363133, counted by A363132.
For maximum instead of minimum we have A363218, counted by A237753.
A055396 gives minimum prime index, maximum A061395.
A112798 lists prime indices, length A001222, sum A056239.
A360005 gives twice median of prime indices.
Cf. A000961, A006141, A046660, A051293, A106529, A111907, A237755, A237824, A327482, A361860, A361861, A362050.

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

A001222(a(n)) = 2*A055396(a(n)).

A363218 Positive integers whose prime indices satisfy: (length) = 2*(maximum).

Original entry on oeis.org

4, 24, 36, 54, 81, 160, 240, 360, 400, 540, 600, 810, 896, 900, 1000, 1215, 1344, 1350, 1500, 2016, 2025, 2240, 2250, 2500, 3024, 3136, 3360, 3375, 3750, 4536, 4704, 5040, 5600, 5625, 5632, 6250, 6804, 7056, 7560, 7840, 8400, 8448, 9375, 10206, 10584, 10976

Offset: 1

Gus Wiseman, May 23 2023

nonn

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 terms together with their prime indices begin:
      4: {1,1}
     24: {1,1,1,2}
     36: {1,1,2,2}
     54: {1,2,2,2}
     81: {2,2,2,2}
    160: {1,1,1,1,1,3}
    240: {1,1,1,1,2,3}
    360: {1,1,1,2,2,3}
    400: {1,1,1,1,3,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    810: {1,2,2,2,2,3}
    896: {1,1,1,1,1,1,1,4}
    900: {1,1,2,2,3,3}
   1000: {1,1,1,3,3,3}
   1215: {2,2,2,2,2,3}
   1344: {1,1,1,1,1,1,2,4}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   2016: {1,1,1,1,1,2,2,4}
   2025: {2,2,2,2,3,3}
   2240: {1,1,1,1,1,1,3,4}

The LHS (number of prime indices) is A001222.
The RHS is twice A061395.
Before multiplying by 2 we had A106529.
Partitions of this type are counted by A237753.
For sum instead of length we have A344415, counted by A035363.
An adjoint version is A361909, also counted by A237753.
For minimum instead of maximum we have A363134, counted by A237757.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
Cf. A001221, A067801, A118096, A324521, A237752, A361855, A361908, A363222.

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

Disjoint from A361909.

A362047 Numbers whose prime indices satisfy: (maximum) - (minimum) = (mean).

Original entry on oeis.org

10, 30, 39, 90, 98, 99, 100, 115, 259, 270, 273, 300, 490, 495, 517, 663, 665, 793, 810, 900, 1000, 1083, 1241, 1421, 1495, 1521, 1691, 1911, 2058, 2079, 2125, 2145, 2369, 2430, 2450, 2475, 2662, 2700, 2755, 2821, 3000, 3277, 4247, 4495, 4921, 5587, 5863, 6069

Offset: 1

Gus Wiseman, Apr 11 2023

nonn

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 terms together with their prime indices begin:
      10: {1,3}
      30: {1,2,3}
      39: {2,6}
      90: {1,2,2,3}
      98: {1,4,4}
      99: {2,2,5}
     100: {1,1,3,3}
     115: {3,9}
     259: {4,12}
     270: {1,2,2,2,3}
     273: {2,4,6}
     300: {1,1,2,3,3}
The prime indices of 490 are {1,3,4,4}, with minimum 1, maximum 4, and mean 3, and 4-1 = 3, so 490 is in the sequence.

Partitions of this type are counted by A361862.
For minimum instead of mean we have A361908, counted by A118096.
A055396 gives minimum prime index, A061395 maximum.
A112798 list prime indices, length A001222, sum A056239.
A243055 subtracts the least prime index from the greatest.
A326844 gives the diagram complement size of Heinz partition.
Cf. A111907, A237832, A268192, A316413, A326836, A326837, A326846, A359358, A359360, A361855.

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

from itertools import count, islice
from sympy import primepi, factorint
def A362047_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(primepi(max(f:=factorint(n)))-primepi(min(f)))*sum(f.values())==sum(primepi(i)*j for i, j in f.items()),count(max(startvalue,2)))
A362047_list = list(islice(A362047_gen(),20)) # Chai Wah Wu, Apr 13 2023

A359360(a(n)) = A326844(a(n)).
A243055(a(n)) = A061395(a(n)) - A055396(a(n))
              = A326567(a(n))/A326568(a(n))
              = A056239(a(n))/A001222(a(n)).

A363133 Numbers > 1 whose prime indices satisfy 2*(minimum) = (mean).

Original entry on oeis.org

10, 28, 30, 39, 84, 88, 90, 100, 115, 171, 208, 252, 255, 259, 264, 270, 273, 280, 300, 363, 517, 544, 624, 756, 783, 784, 792, 793, 810, 840, 880, 900, 925, 1000, 1035, 1085, 1197, 1216, 1241, 1425, 1495, 1521, 1595, 1615, 1632, 1683, 1691, 1785, 1872, 1911

Offset: 1

Gus Wiseman, May 29 2023

nonn

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 terms together with their prime indices begin:
    10: {1,3}
    28: {1,1,4}
    30: {1,2,3}
    39: {2,6}
    84: {1,1,2,4}
    88: {1,1,1,5}
    90: {1,2,2,3}
   100: {1,1,3,3}
   115: {3,9}
   171: {2,2,8}
   208: {1,1,1,1,6}
   252: {1,1,2,2,4}
   255: {2,3,7}
   259: {4,12}
   264: {1,1,1,2,5}

Removing the factor 2 gives A000961.
For maximum instead of mean we have A361908, counted by A118096.
Partitions of this type are counted by A363132.
For length instead of mean we have A363134, counted by A237757.
For 2*(maximum) = (length) we have A363218, counted by A237753.
A051293 counts subsets with integer mean.
A112798 lists prime indices, length A001222, sum A056239.
A360005 gives twice median of prime indices.
Cf. A006141, A106529, A111907, A237755, A237824, A324522, A327482, A361860, A361861, A362050.

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

A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 3, 2, 2, 0, 7, 0, 3, 6, 10, 0, 13, 0, 17, 10, 5, 0, 40, 12, 6, 18, 34, 0, 62, 0, 50, 24, 8, 60, 125, 0, 9, 32, 169, 0, 165, 0, 95, 176, 11, 0, 373, 114, 198, 54, 143, 0, 384, 254, 574, 66, 14, 0, 1090, 0, 15, 748, 633, 448, 782, 0, 286

Offset: 1

Gus Wiseman, Apr 10 2023

nonn

In terms of partition diagrams, these are partitions whose rectangle from the left (length times minimum) has the same size as the complement.

			The a(4) = 1 through a(12) = 7 partitions:
  (31)  .  (321)  .  (62)    (441)  (32221)  .  (93)
                     (3221)  (522)  (33211)     (642)
                     (3311)                     (4431)
                                                (5322)
                                                (322221)
                                                (332211)
                                                (333111)
The partition y = (4,4,3,1) has maximum 4 and minimum 1 and mean 3, and 4 - 1 = 3, so y is counted under a(12). The diagram of y is:
  o o o o
  o o o o
  o o o .
  o . . .
Both the rectangle from the left and the complement have size 4.

Positions of zeros are 1 and A000040.
For length instead of mean we have A237832.
For minimum instead of mean we have A118096.
These partitions have ranks A362047.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A067538 counts partitions with integer mean.
A097364 counts partitions by (maximum) - (minimum).
A243055 subtracts the least prime index from the greatest.
A326844 gives the diagram complement size of Heinz partition.
Cf. A237984, A240219, A326836, A326837, A327482, A237755, A237824, A349156, A359360, A360068, A360241, A361853.

Table[Length[Select[IntegerPartitions[n],Max@@#-Min@@#==Mean[#]&]],{n,30}]

cp's OEIS Frontend

A239500 Number of partitions p of n such that if h = (number of parts of p), then h is an (h,1)-separator of p; see Comments.

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A239501 Number of partitions p of n such that if h = max(p) - min(p), then h is an (h,1)-separator of p; see Comments.

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A361854 Number of strict integer partitions of n such that (length) * (maximum) = 2n.

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A361861 Number of integer partitions of n where the median is twice the minimum.

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

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A363134 Positive integers whose multiset of prime indices satisfies: (length) = 2*(minimum).

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A363218 Positive integers whose prime indices satisfy: (length) = 2*(maximum).

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A362047 Numbers whose prime indices satisfy: (maximum) - (minimum) = (mean).

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A363133 Numbers > 1 whose prime indices satisfy 2*(minimum) = (mean).