A326849 -id:A326849 - cp's OEIS frontend

A349156 Number of integer partitions of n whose mean is not an integer.

1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752

Offset: 0

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

By conjugation, also the number of integer partitions of n with greatest part not dividing n.

			The a(3) = 1 through a(8) = 11 partitions:
  (21)  (211)  (32)    (2211)   (43)      (332)
               (41)    (3111)   (52)      (422)
               (221)   (21111)  (61)      (431)
               (311)            (322)     (521)
               (2111)           (331)     (611)
                                (421)     (22211)
                                (511)     (32111)
                                (2221)    (41111)
                                (3211)    (221111)
                                (4111)    (311111)
                                (22111)   (2111111)
                                (31111)
                                (211111)

Below, "!" means either enumerative or set theoretical complement.
The version for nonempty subsets is !A051293.
The complement is counted by A067538, ranked by A316413.
The geometric version is !A067539, strict !A326625, ranked by !A326623.
The strict case is !A102627.
The version for prime factors is A175352, complement A078175.
The version for distinct prime factors is A176587, complement A078174.
The ordered version (compositions) is !A271654, ranked by !A096199.
The multiplicative version (factorizations) is !A326622, geometric !A326028.
The conjugate is ranked by !A326836.
The conjugate strict version is !A326850.
These partitions are ranked by A348551.
A000041 counts integer partitions.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A236634 counts unbalanced partitions, complement of A047993.
A327472 counts partitions not containing their mean, complement of A237984.
Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.

Table[Length[Select[IntegerPartitions[n],!IntegerQ[Mean[#]]&]],{n,0,30}]

a(n > 0) = A000041(n) - A067538(n).

A340610 Numbers whose number of prime factors (A001222) divides their greatest prime index (A061395).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 11, 13, 14, 17, 19, 20, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 45, 47, 49, 50, 52, 53, 56, 57, 58, 59, 61, 65, 67, 71, 73, 74, 75, 78, 79, 83, 84, 86, 87, 89, 91, 92, 95, 97, 101, 103, 106, 107, 109, 111, 113, 117, 122, 125, 126, 127

Offset: 1

Gus Wiseman, Jan 27 2021

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 sequence of terms together with their prime indices begins:
     2: {1}        29: {10}       56: {1,1,1,4}
     3: {2}        30: {1,2,3}    57: {2,8}
     5: {3}        31: {11}       58: {1,10}
     6: {1,2}      35: {3,4}      59: {17}
     7: {4}        37: {12}       61: {18}
     9: {2,2}      38: {1,8}      65: {3,6}
    11: {5}        39: {2,6}      67: {19}
    13: {6}        41: {13}       71: {20}
    14: {1,4}      43: {14}       73: {21}
    17: {7}        45: {2,2,3}    74: {1,12}
    19: {8}        47: {15}       75: {2,3,3}
    20: {1,1,3}    49: {4,4}      78: {1,2,6}
    21: {2,4}      50: {1,3,3}    79: {22}
    23: {9}        52: {1,1,6}    83: {23}
    26: {1,6}      53: {16}       84: {1,1,2,4}

Robert Israel, Table of n, a(n) for n = 1..10000

Note: Heinz numbers are given in parentheses below.
The case of equality is A047993 (A106529).
The case where all parts are multiples, not just the maximum part, is A143773 (A316428), with strict case A340830, while the case of factorizations is A340853.
These are the Heinz numbers of certain partitions counted by A168659.
The reciprocal version is A340609.
The squarefree case is A340828 (A340856).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413).
A067538 counts partitions whose maximum divides their sum (A326836).
A112798 lists the prime indices of each positive integer.
A200750 counts partitions with length coprime to maximum (A340608).
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
Cf. A244990/A244991, A326837, A326849 (A326848), A340653, A340691, A340693 (A340606), A340787/A340788.

filter:= proc(n) local F,m,g,t;
  F:= ifactors(n)[2];
  m:= add(t[2],t=F);
  g:= numtheory:-pi(max(seq(t[1],t=F)));
  g mod m = 0;
end proc:
select(filter, [$2..1000]); # Robert Israel, Feb 08 2021

Select[Range[2,100],Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

A001222(a(n)) divides A061395(a(n)).

A340609 Numbers whose number of prime factors (A001222) is divisible by their greatest prime index (A061395).

Original entry on oeis.org

2, 4, 6, 8, 9, 16, 20, 24, 30, 32, 36, 45, 50, 54, 56, 64, 75, 81, 84, 96, 125, 126, 128, 140, 144, 160, 176, 189, 196, 210, 216, 240, 256, 264, 294, 315, 324, 350, 360, 384, 396, 400, 416, 440, 441, 486, 490, 512, 525, 540, 576, 594, 600, 616, 624, 660, 686

Offset: 1

Gus Wiseman, Jan 27 2021

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.

If n is a term, then so is n^k for k > 1. - Robert Israel, Feb 08 2021

			The sequence of terms together with their prime indices begins:
      2: {1}             64: {1,1,1,1,1,1}      216: {1,1,1,2,2,2}
      4: {1,1}           75: {2,3,3}            240: {1,1,1,1,2,3}
      6: {1,2}           81: {2,2,2,2}          256: {1,1,1,1,1,1,1,1}
      8: {1,1,1}         84: {1,1,2,4}          264: {1,1,1,2,5}
      9: {2,2}           96: {1,1,1,1,1,2}      294: {1,2,4,4}
     16: {1,1,1,1}      125: {3,3,3}            315: {2,2,3,4}
     20: {1,1,3}        126: {1,2,2,4}          324: {1,1,2,2,2,2}
     24: {1,1,1,2}      128: {1,1,1,1,1,1,1}    350: {1,3,3,4}
     30: {1,2,3}        140: {1,1,3,4}          360: {1,1,1,2,2,3}
     32: {1,1,1,1,1}    144: {1,1,1,1,2,2}      384: {1,1,1,1,1,1,1,2}
     36: {1,1,2,2}      160: {1,1,1,1,1,3}      396: {1,1,2,2,5}
     45: {2,2,3}        176: {1,1,1,1,5}        400: {1,1,1,1,3,3}
     50: {1,3,3}        189: {2,2,2,4}          416: {1,1,1,1,1,6}
     54: {1,2,2,2}      196: {1,1,4,4}          440: {1,1,1,3,5}
     56: {1,1,1,4}      210: {1,2,3,4}          441: {2,2,4,4}

Robert Israel, Table of n, a(n) for n = 1..10000

Note: Heinz numbers are given in parentheses below.
The case of equality is A047993 (A106529).
These are the Heinz numbers of certain partitions counted by A168659.
The reciprocal version is A340610, with strict case A340828 (A340856).
If all parts (not just the greatest) are divisors we get A340693 (A340606).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413).
A067538 counts partitions whose maximum divides their sum (A326836).
A112798 lists the prime indices of each positive integer.
A200750 counts partitions with length coprime to maximum (A340608).
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
Cf. A039900, A064174, A143773, A244990/A244991, A326837, A326849 (A326848), A340653, A340787/A340788.

filter:= proc(n) local F,m,g,t;
  F:= ifactors(n)[2];
  m:= add(t[2],t=F);
  g:= numtheory:-pi(max(seq(t[1],t=F)));
  m mod g = 0;
end proc:
seelect(filter, [$2..1000]); # Robert Israel, Feb 08 2021

Select[Range[2,100],Divisible[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]&]

A061395(a(n)) divides A001222(a(n)).

A326843 Number of integer partitions of n whose length and maximum both divide n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 22, 2, 5, 11, 16, 2, 36, 2, 46, 22, 5, 2, 209, 3, 5, 42, 130, 2, 434, 2, 217, 77, 5, 52, 1400, 2, 5, 135, 1749, 2, 1782, 2, 957, 2151, 5, 2, 8355, 3, 1859, 385, 2388, 2, 6726, 2765, 10641, 627, 5, 2, 68049, 2, 5, 13424, 17142

Offset: 0

Gus Wiseman, Jul 26 2019

nonn

The Heinz numbers of these partitions are given by A326837.

			The a(1) = 1 through a(8) = 5 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (1111)           (222)                (2222)
                                     (321)                (4211)
                                     (111111)             (11111111)
The a(12) = 22 partitions:
  (12)
  (6,6)
  (4,4,4)
  (6,3,3)
  (6,4,2)
  (6,5,1)
  (3,3,3,3)
  (4,3,3,2)
  (4,4,2,2)
  (4,4,3,1)
  (6,2,2,2)
  (6,3,2,1)
  (6,4,1,1)
  (2,2,2,2,2,2)
  (3,2,2,2,2,1)
  (3,3,2,2,1,1)
  (3,3,3,1,1,1)
  (4,2,2,2,1,1)
  (4,3,2,1,1,1)
  (4,4,1,1,1,1)
  (6,2,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..180

The strict case is A326851.
The non-constant case is A326852.
The case where all parts (not just the maximum) divide n is A326842.
Cf. A018818, A047993, A067538, A326837, A326849.

Table[If[n==0,1,Length[Select[IntegerPartitions[n],Divisible[n,Length[#]]&&Divisible[n,Max[#]]&]]],{n,0,30}]

A340608 The number of prime factors of n (A001222) is relatively prime to the maximum prime index of n (A061395).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 22, 23, 25, 27, 28, 29, 31, 32, 33, 34, 37, 40, 41, 42, 43, 44, 46, 47, 48, 51, 53, 55, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 76, 77, 79, 80, 82, 83, 85, 88, 89, 90, 93, 94, 97, 98, 99

Offset: 1

Gus Wiseman, Jan 27 2021

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 sequence of terms together with their prime indices begins:
     2: {1}          22: {1,5}          44: {1,1,5}
     3: {2}          23: {9}            46: {1,9}
     4: {1,1}        25: {3,3}          47: {15}
     5: {3}          27: {2,2,2}        48: {1,1,1,1,2}
     7: {4}          28: {1,1,4}        51: {2,7}
     8: {1,1,1}      29: {10}           53: {16}
    10: {1,3}        31: {11}           55: {3,5}
    11: {5}          32: {1,1,1,1,1}    59: {17}
    12: {1,1,2}      33: {2,5}          60: {1,1,2,3}
    13: {6}          34: {1,7}          61: {18}
    15: {2,3}        37: {12}           62: {1,11}
    16: {1,1,1,1}    40: {1,1,1,3}      63: {2,2,4}
    17: {7}          41: {13}           64: {1,1,1,1,1,1}
    18: {1,2,2}      42: {1,2,4}        66: {1,2,5}
    19: {8}          43: {14}           67: {19}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Note: Heinz numbers are given in parentheses below.
These are the Heinz numbers of the partitions counted by A200750.
The case of equality is A047993 (A106529).
The divisible instead of coprime version is A168659 (A340609).
The dividing instead of coprime version is A168659 (A340610), with strict case A340828 (A340856).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A051424 counts singleton or pairwise coprime partitions (A302569).
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413).
A067538 counts partitions whose maximum divides their sum (A326836).
A112798 lists the prime indices of each positive integer.
A259936 counts singleton or pairwise coprime factorizations.
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
A326849 counts partitions whose sum divides length times maximum (A326848).
A327516 counts pairwise coprime partitions (A302696).
Cf. A003114, A039900, A096401, A244990/A244991, A340606, A340653, A340691, A340787/A340788.

Select[Range[100],GCD[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]==1&]

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

Gus Wiseman, Mar 29 2023

nonn

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

These are partitions whose diagram has the same size as its complement (see example).

			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).

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.

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

A326851 Number of strict integer partitions of n whose length and maximum both divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 5, 1, 6, 1, 1, 1, 16, 1, 1, 1, 12, 1, 33, 1, 15, 1, 1, 1, 60, 1, 1, 1, 51, 1, 81, 1, 31, 57, 1, 1, 216, 1, 55, 1, 45, 1, 230, 1, 223, 1, 1, 1, 800, 1, 1, 314, 273, 1, 607, 1, 81, 1, 315, 1, 2404, 1, 1, 319

Offset: 0

Gus Wiseman, Jul 26 2019

nonn

			The a(6) = 2 through a(24) = 16 partitions (1 terms not shown):
  6       12        15          16        18      20           24
  3,2,1   6,4,2     5,4,3,2,1   8,4,3,1   9,5,4   10,5,3,2     12,7,5
          6,5,1                 8,5,2,1   9,6,3   10,5,4,1     12,8,4
          6,3,2,1                         9,7,2   10,6,3,1     12,9,3
                                          9,8,1   10,7,2,1     12,10,2
                                                  10,4,3,2,1   12,11,1
                                                               8,7,5,4
                                                               8,7,6,3
                                                               12,5,4,3
                                                               12,6,4,2
                                                               12,6,5,1
                                                               12,7,3,2
                                                               12,7,4,1
                                                               12,8,3,1
                                                               12,9,2,1
                                                               8,6,4,3,2,1

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..383

The non-strict case is A326843.

Cf. A018818, A047993, A067538, A326837, A326842, A326849, A326852.

Table[If[n==0,1,Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[n,Max[#]]&&Divisible[n,Length[#]]&]]],{n,0,30}]

A326848 Heinz numbers of integer partitions of m >= 0 whose length times maximum is a multiple of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 78, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 171, 173, 179, 181

Offset: 1

Gus Wiseman, Jul 26 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A326849.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Cf. A001222, A047993, A056239, A061395, A067538, A112798, A316413, A326836, A326843, A326847, A326849, A326851.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],#==1||Divisible[Max[primeMS[#]]*Length[primeMS[#]],Total[primeMS[#]]]&]

A268192 Triangle read by rows: T(n,k) is the number of partitions of weight k among the complements of the partitions of n.

Original entry on oeis.org

1, 2, 2, 1, 3, 0, 2, 2, 2, 0, 2, 1, 4, 0, 2, 1, 2, 0, 2, 2, 2, 2, 2, 0, 4, 0, 0, 2, 1, 4, 1, 2, 0, 6, 0, 2, 2, 1, 0, 2, 0, 2, 3, 2, 0, 6, 0, 2, 4, 4, 0, 2, 0, 2, 2, 0, 0, 2, 1, 4, 0, 6, 0, 2, 4, 5, 0, 6, 0, 4, 2, 0, 0, 4, 1, 0, 0, 2, 0, 2, 2, 4, 0, 2, 6, 5, 0, 6, 0, 8

Offset: 1

Emeric Deutsch, Feb 12 2016

The complement of a partition p[1] >= p[2] >=...>= p[k] is p[1]-p[2], p[1]-p[3], ..., p[1]-p[k]. Its Ferrers board emerges naturally from the Ferrers board of the given partition. The weight of a partition of n is n.
Sum of entries in row n is A000041(n) (the partition numbers).
Apparently, number of entries in row n is A033638(n-1) = 1 + floor((n-1)^2/4).
T(n,0) = A000005(n) = number of divisors of n.
T(n,1) = A070824(n+1).
Sum(k*T(n,k),k>0) = A188814(n).

			Row 4 is 3,0,2; indeed, the complements of [4], [3,1], [2,2], [2,1,1], [1,1,1,1] are: empty, [2], empty, [1,1], empty; their weights are 0, 2, 0, 2, 0, respectively.
From _Gus Wiseman_, Sep 24 2019: (Start)
Triangle begins:
  1
  2
  2 1
  3 0 2
  2 2 0 2 1
  4 0 2 1 2 0 2
  2 2 2 2 0 4 0 0 2 1
  4 1 2 0 6 0 2 2 1 0 2 0 2
  3 2 0 6 0 2 4 4 0 2 0 2 2 0 0 2 1
  4 0 6 0 2 4 5 0 6 0 4 2 0 0 4 1 0 0 2 0 2
  2 4 0 2 6 5 0 6 0 8 4 0 0 6 2 0 2 2 0 2 0 2 0 0 2 1
Row  n = 8 counts the following partitions:
  8          332   53      62       71        521     4211   611      5111
  44               22211   422      2111111   32111          311111   41111
  2222                     431
  11111111                 3221
                           3311
                           221111
(End)

Alois P. Heinz, Rows n = 1..70, flattened

Cf. A000005, A000041, A002620, A033638, A070824, A188814, A326844, A326849.

q := 10: with(combinat): a := proc (i, j) options operator, arrow: partition(i)[j] end proc: P[q] := 0: for j to numbpart(q) do P[q] := sort(P[q]+t^(nops(a(q, j))*max(a(q, j))-q)) end do: P[q] := P[q];
# second Maple program:
b:= proc(n, i, l) option remember; expand(`if`(n=0 or i=1,
      x^(`if`(l=0, 0, n*(l-i))), b(n, i-1, l)+`if`(i>n, 0,
      x^(`if`(l=0, 0, l-i))*b(n-i, i, `if`(l=0, i, l)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=1..15);  # Alois P. Heinz, Feb 12 2016

b[n_, i_, l_] := b[n, i, l] = Expand[If[n == 0 || i == 1, x^(If[l == 0, 0, n*(l - i)]), b[n, i - 1, l] + If[i > n, 0, x^(If[l == 0, 0, l - i])*b[n - i, i, If[l == 0, i, l]]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Max[#]*Length[#]-n==k&]],{n,1,11},{k,0,Floor[(n-1)/2]*Ceiling[(n-1)/2]}] (* Gus Wiseman, Sep 24 2019 *)

The weight of the complement of a partition p is (number of parts of p)*(largest part of p) - weight of p.

For a given q, the Maple program yields the generating polynomial of row q.

A340830 Number of strict integer partitions of n such that every part is a multiple of the number of parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 1, 4, 1, 6, 1, 5, 2, 6, 1, 8, 1, 7, 4, 7, 1, 12, 1, 8, 6, 9, 1, 16, 1, 10, 9, 11, 1, 21, 1, 12, 13, 12, 1, 28, 1, 13, 17, 16, 1, 33, 1, 19, 22, 15, 1, 45, 1, 16, 28, 25, 1, 47, 1, 28, 34, 18

Offset: 1

Gus Wiseman, Feb 02 2021

nonn

			The a(n) partitions for n = 1, 6, 10, 14, 18, 20, 24, 26, 30:
  1   6     10    14     18      20     24       26      30
      4,2   6,4   8,6    10,8    12,8   16,8     18,8    22,8
            8,2   10,4   12,6    14,6   18,6     20,6    24,6
                  12,2   14,4    16,4   20,4     22,4    26,4
                         16,2    18,2   22,2     24,2    28,2
                         9,6,3          14,10    14,12   16,14
                                        12,9,3   16,10   18,12
                                        15,6,3           20,10
                                                         15,9,6
                                                         18,9,3
                                                         21,6,3
                                                         15,12,3

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict case is A143773 (A316428).
The case where length divides sum also is A340827.
The version for factorizations is A340851.
Factorization of this type are counted by A340853.
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions whose length/max divide sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A102627 counts strict partitions whose length divides sum.
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340828 counts strict partitions with length divisible by maximum.
A340829 counts strict partitions with Heinz number divisible by sum.
Cf. A114638, A168659, A326641, A326843 (A326837), A326849, A326852 (A326838), A330950 (A324851), A340852.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@IntegerQ/@(#/Length[#])&]],{n,30}]

a(n) = Sum_{d|n} A008289(n/d, d).

cp's OEIS Frontend

A349156 Number of integer partitions of n whose mean is not an integer.

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A340610 Numbers whose number of prime factors (A001222) divides their greatest prime index (A061395).

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A340609 Numbers whose number of prime factors (A001222) is divisible by their greatest prime index (A061395).

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A326843 Number of integer partitions of n whose length and maximum both divide n.

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A340608 The number of prime factors of n (A001222) is relatively prime to the maximum prime index of n (A061395).

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

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A326851 Number of strict integer partitions of n whose length and maximum both divide n.

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A326848 Heinz numbers of integer partitions of m >= 0 whose length times maximum is a multiple of m.

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