A091602 -id:A091602 - cp's OEIS frontend

A382302 Number of integer partitions of n with greatest part, greatest multiplicity, and number of distinct parts all equal.

0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 2, 2, 2, 4, 3, 3, 4, 4, 3, 6, 5, 8, 8, 13, 13, 16, 17, 21, 22, 25, 26, 32, 34, 37, 44, 47, 55, 62, 72, 78, 94, 103, 118, 132, 151, 163, 189, 205, 230, 251, 284, 307, 346, 377, 420, 462, 515, 562, 629, 690, 763

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(n) partitions for n = 1, 2, 10, 13, 14, 19, 20, 21:
  1  .  32221   332221   333221   4333321     43333211    43333221
        322111  333211   3322211  43322221    44322221    433332111
                3322111  3332111  433321111   433222211   443222211
                4321111           443221111   443321111   444321111
                                  543211111   4332221111  4332222111
                                  4322221111              4333221111
                                                          4432221111
                                                          5432211111

John Tyler Rascoe, Table of n, a(n) for n = 0..500

Without the middle statistic we have A000009, ranked by A055932.
Counting partitions by the LHS gives A008284 (strict A008289), rank statistic A061395.
Counting partitions by the middle statistic gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116608/A365676, rank statistic A001221.
Without the LHS we have A239964, ranked by A212166.
Without the RHS we have A240312, ranked by A381542.
The Heinz numbers of these partitions are listed by A381543.
A000041 counts integer partitions.
A047993 counts partitions with max part = length, ranks A106529.
A116598 counts ones in partitions, rank statistic A007814.
A381438 counts partitions by last part part of section-sum partition.
Cf. A047966, A130091, A237984, A239455, A241131, A351293, A362608, A363719, A381079, A381544, A382303.

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

A_x(N) = {if(N<1,[0],my(x='x+O('x^(N+1))); concat([0],Vec(sum(i=1,N, prod(j=1,i, (x^j-x^((i+1)*j))/(1-x^j)) - prod(j=1,i, (x^j-x^(i*j))/(1-x^j))))))}
A_x(60) \\ John Tyler Rascoe, Mar 25 2025

G.f.: Sum_{i>0} (B(i+1,i,x) - B(i,i,x)) where B(a,c,x) = Product_{j=1..c} (x^j - x^(a*j))/(1 - x^j). - John Tyler Rascoe, Mar 25 2025

A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

Original entry on oeis.org

2, 12, 18, 36, 120, 270, 360, 540, 600, 750, 1080, 1350, 1500, 1680, 1800, 2250, 2700, 3000, 4500, 5040, 5400, 5670, 6750, 8400, 9000, 11340, 11760, 13500, 15120, 22680, 25200, 26250, 27000, 28350, 35280, 36960, 39690, 42000, 45360, 52500, 56700, 58800, 72030

Offset: 1

Gus Wiseman, Mar 24 2025

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, sum A056239.

			The terms together with their prime indices begin:
      2: {1}
     12: {1,1,2}
     18: {1,2,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    270: {1,2,2,2,3}
    360: {1,1,1,2,2,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    750: {1,2,3,3,3}
   1080: {1,1,1,2,2,2,3}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   1680: {1,1,1,1,2,3,4}
   1800: {1,1,1,2,2,3,3}

Counting partitions by the LHS gives A008284, rank statistic A061395.
Without the RHS we have A055932, counted by A000009.
Counting partitions by the RHS gives A091602, rank statistic A051903.
Counting partitions by the middle statistic gives A116608/A365676, rank stat A001221.
Without the LHS we have A212166, counted by A239964.
Without the middle statistic we have A381542, counted by A240312.
Partitions of this type are counted by A382302.
A000040 lists the primes, differences A001223.
A001222 counts prime factors, distinct A001221.
A047993 counts balanced partitions, ranks A106529.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents partition conjugation in terms of Heinz numbers.
Cf. A000720, A048767, A062457, A066328, A130091, A317090, A363719, A363740, A380955, A381632.

Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==PrimeNu[#]==Max@@FactorInteger[#][[All,2]]&]

A061395(a(n)) = A001221(a(n)) = A051903(a(n)).

A381079 Number of integer partitions of n whose greatest multiplicity is equal to their sum of distinct parts.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 0, 3, 1, 3, 1, 2, 0, 7, 2, 6, 7, 11, 3, 19, 8, 22, 16, 32, 17, 48, 21, 50, 39, 71, 35, 101, 58, 120, 89, 156, 97, 228, 133, 267, 203, 352, 228, 483, 322, 571, 444, 734, 524, 989, 683, 1160, 942, 1490, 1103, 1919, 1438, 2302, 1890, 2881, 2243, 3683, 2842, 4384, 3703, 5461

Offset: 0

Gus Wiseman, Mar 03 2025

nonn

Are there only 4 zeros?

			The partition (3,2,2,1,1,1,1,1,1) has greatest multiplicity 6 and distinct parts (3,2,1) with sum 6, so is counted under a(13).
The a(1) = 1 through a(13) = 7 partitions:
  1  .  .  22  2111  .  2221   22211  333     331111  5111111   .  33331
                        22111         222111          32111111     322222
                        31111         411111                       3331111
                                                                   4411111
                                                                   61111111
                                                                   322111111
                                                                   421111111

For greatest part instead of multiplicity we have A000005.
Counting partitions by the LHS gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116861, rank statistic A066328.
These partitions are ranked by A381632, for part instead of multiplicity A246655.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts balanced partitions, ranks A106529.
A091605 counts partitions with greatest multiplicity 2.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
Cf. A027193, A047966, A048767, A051904, A212166, A237984, A239455, A241131, A362608, A363724.

Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]==Total[Union[#]]&]],{n,0,30}]

A381544 Number of integer partitions of n not containing more ones than any other part.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 7, 8, 13, 17, 24, 30, 45, 54, 75, 97, 127, 160, 212, 263, 342, 427, 541, 672, 851, 1046, 1307, 1607, 1989, 2428, 2993, 3631, 4443, 5378, 6533, 7873, 9527, 11424, 13752, 16447, 19701, 23470, 28016, 33253, 39537, 46801, 55428, 65408, 77238

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(2) = 1 through a(9) = 17 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)      (9)
       (21)  (22)  (32)   (33)    (43)    (44)     (54)
             (31)  (41)   (42)    (52)    (53)     (63)
                   (221)  (51)    (61)    (62)     (72)
                          (222)   (322)   (71)     (81)
                          (321)   (331)   (332)    (333)
                          (2211)  (421)   (422)    (432)
                                  (2221)  (431)    (441)
                                          (521)    (522)
                                          (2222)   (531)
                                          (3221)   (621)
                                          (3311)   (3222)
                                          (22211)  (3321)
                                                   (4221)
                                                   (22221)
                                                   (32211)
                                                   (222111)

The complement is counted by A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A381439.
The case of equality is A382303, ranks A360014.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max part = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A091605, A116861, A232697, A237984, A362608, A363724, A381079, A381437, A381438.

Table[Length[Select[IntegerPartitions[n],Count[#,1]<=Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

A354234 Triangle read by rows where T(n,k) is the number of integer partitions of n with at least one part divisible by k.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 3, 1, 1, 7, 4, 2, 1, 1, 11, 7, 4, 2, 1, 1, 15, 10, 6, 3, 2, 1, 1, 22, 16, 9, 6, 3, 2, 1, 1, 30, 22, 14, 8, 5, 3, 2, 1, 1, 42, 32, 20, 13, 8, 5, 3, 2, 1, 1, 56, 44, 29, 18, 12, 7, 5, 3, 2, 1, 1, 77, 62, 41, 27, 17, 12, 7, 5, 3, 2, 1, 1

Offset: 1

Gus Wiseman, May 22 2022

Also partitions of n with at least one part appearing k or more times. It would be interesting to have a bijective proof of this.

			Triangle begins:
   1
   2  1
   3  1  1
   5  3  1  1
   7  4  2  1  1
  11  7  4  2  1  1
  15 10  6  3  2  1  1
  22 16  9  6  3  2  1  1
  30 22 14  8  5  3  2  1  1
  42 32 20 13  8  5  3  2  1  1
  56 44 29 18 12  7  5  3  2  1  1
  77 62 41 27 17 12  7  5  3  2  1  1
For example, row n = 5 counts the following partitions:
  (5)      (32)    (32)   (41)  (5)
  (32)     (41)    (311)
  (41)     (221)
  (221)    (2111)
  (311)
  (2111)
  (11111)
At least one part appearing k or more times:
  (5)      (221)    (2111)   (11111)  (11111)
  (32)     (311)    (11111)
  (41)     (2111)
  (221)    (11111)
  (311)
  (2111)
  (11111)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

The complement is counted by A061199.
Differences of consecutive terms are A091602.
Column k = 1 is A000041.
Column k = 2 is A047967, ranked by A013929 and A324929.
Column k = 3 is A295341, ranked by A046099 and A354235.
Column k = 4 is A295342.
A000041 counts integer partitions, strict A000009.
A047966 counts uniform partitions.
Cf. A002033, A006918, A064410, A117485, A238394, A238395, A325534.

Table[Length[Select[IntegerPartitions[n],MemberQ[#/k,_?IntegerQ]&]],{n,1,15},{k,1,n}]
- or -
Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]>=k&]],{n,1,15},{k,1,n}]

\\ here P(k,n) is partitions with no part divisible by k as g.f.
P(k,n)={1/prod(i=1, n, 1 - if(i%k, x^i) + O(x*x^n))}
M(n,m=n)={my(p=P(n+1,n)); Mat(vector(m, k, Col(p-P(k,n), -n) ))}
{ my(A=M(12)); for(n=1, #A, print(A[n,1..n])) } \\ Andrew Howroyd, Jan 19 2023

A367588 Number of integer partitions of n with exactly two distinct parts, both appearing with the same multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 5, 9, 6, 9, 10, 11, 8, 15, 9, 16, 14, 15, 11, 23, 14, 18, 18, 23, 14, 30, 15, 26, 22, 24, 22, 38, 18, 27, 26, 38, 20, 42, 21, 37, 36, 33, 23, 53, 27, 42, 34, 44, 26, 54, 34, 53, 38, 42, 29, 74, 30, 45, 49, 57, 40, 66, 33, 58, 46

Offset: 0

Gus Wiseman, Dec 01 2023

nonn

The Heinz numbers of these partitions are given by A268390.

			The a(3) = 1 through a(12) = 9 partitions (A = 10, B = 11):
  (21)  (31)  (32)  (42)    (43)  (53)    (54)      (64)    (65)  (75)
              (41)  (51)    (52)  (62)    (63)      (73)    (74)  (84)
                    (2211)  (61)  (71)    (72)      (82)    (83)  (93)
                                  (3311)  (81)      (91)    (92)  (A2)
                                          (222111)  (3322)  (A1)  (B1)
                                                    (4411)        (4422)
                                                                  (5511)
                                                                  (333111)
                                                                  (22221111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

For any multiplicities we have A002133, ranks A007774.
For any number of distinct parts we have A047966, ranks A072774.
For distinct multiplicities we have A182473, ranks A367589.
These partitions have ranks A268390.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
Cf. A023645, A091602, A116861, A181819, A243978, A367582.

Table[Sum[Floor[(d-1)/2],{d,Divisors[n]}],{n,30}]

G.f.: Sum_{i, j>0} x^(j*(2*i+1))/(1-x^j). - John Tyler Rascoe, Feb 04 2024

A367682 Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.

Original entry on oeis.org

1, 1, 0, 1, 3, 2, 3, 2, 5, 5, 10, 9, 14, 14, 21, 20, 30, 36, 44, 50, 66, 75, 93, 106, 132, 151, 185, 212, 256, 286, 348, 394, 479, 543, 642, 740, 888, 994, 1176, 1350, 1589, 1789, 2109, 2371, 2786, 3144, 3653, 4126, 4811, 5385, 6213

Offset: 0

Gus Wiseman, Nov 30 2023

nonn

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

			The a(1) = 1 through a(10) = 10 partitions:
  (1)  .  (21)  (22)   (41)   (51)    (61)   (71)     (81)    (91)
                (31)   (221)  (321)   (421)  (431)    (333)   (541)
                (211)         (3111)         (521)    (531)   (631)
                                             (3221)   (621)   (721)
                                             (41111)  (4221)  (3322)
                                                              (3331)
                                                              (4321)
                                                              (5221)
                                                              (322111)
                                                              (511111)

The case of strict partitions is A025147, ranks A039956.
The case of distinct multiplicities is A114640, ranks A109297.
These partitions have ranks A367683.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
Cf. A000837, A071625, A072774, A082090, A116861, A367579, A367580, A367581.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Length[Select[IntegerPartitions[n], Sort[Length/@Split[#]]==mmk[#]&]], {n,0,15}]

A264397 Sum of the sizes of the longest clique of all partitions of n.

Original entry on oeis.org

1, 3, 5, 10, 15, 26, 38, 60, 86, 127, 178, 255, 349, 484, 652, 885, 1174, 1565, 2049, 2689, 3481, 4510, 5779, 7407, 9403, 11933, 15029, 18908, 23636, 29511, 36641, 45432, 56063, 69076, 84753, 103833, 126730, 154438, 187584, 227485, 275056, 332066, 399811

Offset: 1

Emeric Deutsch, Nov 20 2015

nonn

All parts of an integer partition with the same value form a clique. The size of a clique is the number of elements in the clique.

a(n) = Sum(k*A091602(n,k), k=1..n).

			a(4) = 10 because the partitions 4,31,22,211,1111 of 4 have longest clique sizes 1,1,2,2,4, respectively.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A091602, A243978.

g := (sum(k*(product(1-x^(j*(k+1)), j = 1 .. 100) - product(1-x^(j*k), j = 1 .. 100)), k = 1 .. 100))/(product(1-x^j, j = 1 .. 100)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 1 .. 50);

from sympy.utilities.iterables import partitions
def A264397(n): return sum(max(p.values()) for p in partitions(n)) # Chai Wah Wu, Sep 17 2023

G.f.: g(x) = sum(k*(product(1-x^{j*(k+1)}, j>=1) - product(1-x^{j*k}, j>=1)), k>=1)/product(1-x^j, j>=1).

A367589 Numbers with exactly two distinct prime factors, both appearing with different exponents.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 224, 232, 236, 242, 244

Offset: 1

Gus Wiseman, Dec 01 2023

nonn

First differs from A177425 in lacking 360.
First differs from A182854 in lacking 360.
These are the Heinz numbers of the partitions counted by A182473.

			The terms together with their prime indices begin:
  12: {1,1,2}
  18: {1,2,2}
  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}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}

The case of any multiplicities is A007774, counts A002133.
These partitions are counted by A182473.
The case of equal exponents is A367590, counts A367588.
A000041 counts integer partitions, strict A000009.
A091602 counts partitions by greatest multiplicity, least A243978.
A098859 counts partitions with distinct multiplicities, ranks A130091.
A116608 counts partitions by number of distinct parts.
Cf. A071625, A072233, A072774, A109297, A367580.

Select[Range[100], PrimeNu[#]==2&&UnsameQ@@Last/@FactorInteger[#]&]

A367590 Numbers with exactly two distinct prime factors, both appearing with the same exponent.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194

Offset: 1

Gus Wiseman, Dec 01 2023

nonn

First differs from A268390 in lacking 210.
First differs from A238748 in lacking 210.
These are the Heinz numbers of the partitions counted by A367588.

			The terms together with their prime indices begin:
     6: {1,2}         57: {2,8}        106: {1,16}
    10: {1,3}         58: {1,10}       111: {2,12}
    14: {1,4}         62: {1,11}       115: {3,9}
    15: {2,3}         65: {3,6}        118: {1,17}
    21: {2,4}         69: {2,9}        119: {4,7}
    22: {1,5}         74: {1,12}       122: {1,18}
    26: {1,6}         77: {4,5}        123: {2,13}
    33: {2,5}         82: {1,13}       129: {2,14}
    34: {1,7}         85: {3,7}        133: {4,8}
    35: {3,4}         86: {1,14}       134: {1,19}
    36: {1,1,2,2}     87: {2,10}       141: {2,15}
    38: {1,8}         91: {4,6}        142: {1,20}
    39: {2,6}         93: {2,11}       143: {5,6}
    46: {1,9}         94: {1,15}       145: {3,10}
    51: {2,7}         95: {3,8}        146: {1,21}
    55: {3,5}        100: {1,1,3,3}    155: {3,11}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

The case of any multiplicities is A007774, counts A002133.
Partitions of this type are counted by A367588.
The case of distinct exponents is A367589, counts A182473.
A000041 counts integer partitions, strict A000009.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
Cf. A006881, A039956, A071625, A072233, A072774, A109297, A303661, A367580.

Select[Range[100], SameQ@@Last/@If[#==1, {}, FactorInteger[#]]&&PrimeNu[#]==2&]
Select[Range[200],PrimeNu[#]==2&&Length[Union[FactorInteger[#][[;;,2]]]]==1&] (* Harvey P. Dale, Aug 04 2025 *)

Union of A006881 and A303661. - Michael De Vlieger, Dec 01 2023

cp's OEIS Frontend

A382302 Number of integer partitions of n with greatest part, greatest multiplicity, and number of distinct parts all equal.

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A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

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A381079 Number of integer partitions of n whose greatest multiplicity is equal to their sum of distinct parts.

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A381544 Number of integer partitions of n not containing more ones than any other part.

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A354234 Triangle read by rows where T(n,k) is the number of integer partitions of n with at least one part divisible by k.

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A367588 Number of integer partitions of n with exactly two distinct parts, both appearing with the same multiplicity.

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A367682 Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.

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A264397 Sum of the sizes of the longest clique of all partitions of n.

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