A326843 -id:A326843 - cp's OEIS frontend

A340606 Numbers whose prime indices (A112798) are all divisors of the number of prime factors (A001222).

1, 2, 4, 6, 8, 9, 16, 20, 24, 32, 36, 50, 54, 56, 64, 81, 84, 96, 125, 126, 128, 144, 160, 176, 189, 196, 216, 240, 256, 294, 324, 360, 384, 400, 416, 441, 486, 512, 540, 576, 600, 624, 686, 729, 810, 864, 896, 900, 936, 968, 1000, 1024, 1029, 1040, 1088, 1215

Offset: 1

Gus Wiseman, Jan 24 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:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
   9: {2,2}
  16: {1,1,1,1}
  20: {1,1,3}
  24: {1,1,1,2}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  50: {1,3,3}
  54: {1,2,2,2}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  81: {2,2,2,2}
  84: {1,1,2,4}
  96: {1,1,1,1,1,2}

Note: Heinz numbers are given in parentheses below.
The reciprocal version is A143773 (A316428).
These partitions are counted by A340693.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A003963 multiplies together the prime indices of n.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length divides n (A316413).
A056239 adds up the prime indices of n.
A061395 selects the maximum prime index.
A067538 counts partitions of n whose maximum divides n (A326836).
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 = partitions whose length is divisible by their maximum (A340609).
A168659 = partitions whose maximum is divisible by their length (A340610).
A289509 lists numbers with relatively prime prime indices.
A326842 = partitions of n whose length and parts all divide n (A326847).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A340852 have a factorization with factors dividing length.
Cf. A000720, A001222, A006141, A067539, A200750, A298423, A324925, A326149/A326155, A340608, A340827.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@IntegerQ/@(PrimeOmega[#]/primeMS[#])&]

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).

A271654 a(n) = Sum_{k|n} binomial(n-1,k-1).

Original entry on oeis.org

1, 2, 2, 5, 2, 17, 2, 44, 30, 137, 2, 695, 2, 1731, 1094, 6907, 2, 30653, 2, 97244, 38952, 352739, 2, 1632933, 10628, 5200327, 1562602, 20357264, 2, 87716708, 2, 303174298, 64512738, 1166803145, 1391282, 4978661179, 2, 17672631939, 2707475853, 69150651910, 2, 286754260229, 2, 1053966829029, 115133177854, 4116715363847, 2, 16892899722499, 12271514, 63207357886437

Offset: 1

Franklin T. Adams-Watters, Apr 11 2016

nonn

Also the number of compositions of n whose length divides n, i.e., compositions with integer mean, ranked by A096199. - Gus Wiseman, Sep 28 2022

			From _Gus Wiseman_, Sep 28 2022: (Start)
The a(1) = 1 through a(6) = 17 compositions with integer mean:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,1,1)  (1,3)      (1,1,1,1,1)  (1,5)
                       (2,2)                   (2,4)
                       (3,1)                   (3,3)
                       (1,1,1,1)               (4,2)
                                               (5,1)
                                               (1,1,4)
                                               (1,2,3)
                                               (1,3,2)
                                               (1,4,1)
                                               (2,1,3)
                                               (2,2,2)
                                               (2,3,1)
                                               (3,1,2)
                                               (3,2,1)
                                               (4,1,1)
                                               (1,1,1,1,1,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..3329

Cf. A056045.
The version for nonempty subsets is A051293, geometric A326027.
The version for partitions is A067538, ranked by A316413, strict A102627.
These compositions are ranked by A096199.
The version for factorizations is A326622, geometric A326028.
A011782 counts compositions.
A067539 = partitions w integer geo mean, ranked by A326623, strict A326625.
A100346 counts compositions into divisors, partitions A018818.
Cf. A000041, A078174, A078175, A326567/A326568, A326624, A326641, A326836, A326837, A326843.

a:= n-> add(binomial(n-1, d-1), d=numtheory[divisors](n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 03 2023

Table[Length[Join @@ Permutations/@Select[IntegerPartitions[n],IntegerQ[Mean[#]]&]],{n,15}] (* Gus Wiseman, Sep 28 2022 *)

PARI
```
a(n)=sumdiv(n,k,binomial(n-1,k-1))
```

A340827 Number of strict integer partitions of n into divisors of n whose length also divides n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 18, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 17, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 12, 1, 1, 1, 3, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

The first element not in A326715 that is however a Heinz number of these partitions is 273.

			The a(n) partitions for n = 6, 12, 24, 90, 84:
  6       12        24            90                      84
  3,2,1   6,4,2     12,8,4        45,30,15                42,28,14
          6,3,2,1   12,6,4,2      45,30,9,5,1             42,21,14,7
                    12,8,3,1      45,18,15,9,3            42,28,12,2
                    8,6,4,3,2,1   45,30,10,3,2            42,28,6,4,3,1
                                  45,18,15,10,2           42,28,7,4,2,1
                                  45,30,6,5,3,1           42,14,12,7,6,3
                                  45,30,9,3,2,1           42,21,12,4,3,2
                                  45,15,10,9,6,5          42,21,12,6,2,1
                                  45,18,10,9,5,3          42,21,14,4,2,1
                                  45,18,10,9,6,2          28,21,14,12,6,3
                                  45,18,15,6,5,1          28,21,14,12,7,2
                                  45,18,15,9,2,1          42,21,7,6,4,3,1
                                  30,18,15,10,6,5,3,2,1   42,14,12,7,4,3,2
                                                          42,14,12,7,6,2,1
                                                          28,21,14,12,4,3,2
                                                          28,21,14,12,6,2,1

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2519 terms from Antti Karttunen)
Antti Karttunen, Scheme program for computing this sequence

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict case is A326842 (A326847).
A018818 = partitions using divisors (A326841).
A047993 = balanced partitions (A106529).
A067538 = partitions whose length/maximum divides sum (A316413/A326836).
A072233 = partitions by sum and length, with strict case A008289.
A102627 = strict partitions whose length divides sum.
A326850 = strict partitions whose maximum part divides sum.
A326851 = strict partitions w/ length and max dividing sum.
A340828 = strict partitions w/ length divisible by max.
A340829 = strict partitions w/ Heinz number divisible by sum.
A340830 = strict partitions w/ parts divisible by length.
Cf. A064173, A143773 (A316428), A168659, A326843 (A326837), A326852, A330950 (A324851), A340851, A340853.

Table[Length[Select[IntegerPartitions[n,All,Divisors[n]],UnsameQ@@#&&Divisible[n,Length[#]]&]],{n,30}]

A340827(n, divsleft=List(divisors(n)), rest=n, len=0) = if(rest<=0, !rest && !(n%len), my(s=0, d); forstep(i=#divsleft, 1, -1, d = divsleft[i]; listpop(divsleft,i); if(rest>=d, s += A340827(n, divsleft, rest-d, 1+len))); (s)); \\ Antti Karttunen, Feb 22 2023

;; See the Links-section. - Antti Karttunen, Feb 22 2023

Data section extended up to a(105) by Antti Karttunen, Feb 22 2023

A327778 Number of integer partitions of n whose LCM is a multiple of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 11, 1, 11, 23, 1, 1, 23, 1, 85, 85, 45, 1, 152, 1, 84, 1, 451, 1, 1787, 1, 1, 735, 260, 1925, 1908, 1, 437, 1877, 4623, 1, 14630, 1, 6934, 10519, 1152, 1, 6791, 1, 1817, 10159, 22556, 1, 2819, 47927, 69333, 22010, 4310, 1

Offset: 0

Gus Wiseman, Sep 25 2019

nonn

			The partitions of n = 6, 10, 12, and 15 whose LCM is a multiple of n:
  (6)      (10)         (12)             (15)
  (3,2,1)  (5,3,2)      (5,4,3)          (6,5,4)
           (5,4,1)      (6,4,2)          (7,5,3)
           (5,2,2,1)    (8,3,1)          (9,5,1)
           (5,2,1,1,1)  (4,3,3,2)        (10,3,2)
                        (4,4,3,1)        (5,4,3,3)
                        (6,4,1,1)        (5,5,3,2)
                        (4,3,2,2,1)      (6,5,2,2)
                        (4,3,3,1,1)      (6,5,3,1)
                        (4,3,2,1,1,1)    (10,3,1,1)
                        (4,3,1,1,1,1,1)  (5,3,3,2,2)
                                         (5,3,3,3,1)
                                         (5,4,3,2,1)
                                         (5,5,3,1,1)
                                         (6,5,2,1,1)
                                         (5,3,2,2,2,1)
                                         (5,3,3,2,1,1)
                                         (5,4,3,1,1,1)
                                         (6,5,1,1,1,1)
                                         (5,3,2,2,1,1,1)
                                         (5,3,3,1,1,1,1)
                                         (5,3,2,1,1,1,1,1)
                                         (5,3,1,1,1,1,1,1,1)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The Heinz numbers of these partitions are given by A327783.
Partitions whose LCM is equal to their sum are A074761.
Partitions whose LCM is greater than their sum are A327779.
Partitions whose LCM is less than their sum are A327781.
Cf. A000961, A018818, A067538, A290103, A319333, A326842, A326843, A327780.

a:= proc(m) option remember; local b; b:=
      proc(n, i, l) option remember; `if`(n=0 or i=1,
        `if`(l=m, 1, 0), `if`(i<2, 0, b(n, i-1, l))+
         b(n-i, min(n-i, i), igcd(m, ilcm(l, i))))
      end; `if`(isprime(m), 1, b(m$2, 1))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 26 2019

Table[Length[Select[IntegerPartitions[n],Divisible[LCM@@#,n]&]],{n,30}]
(* Second program: *)
a[m_] := a[m] = Module[{b}, b[n_, i_, l_] := b[n, i, l] = If[n == 0 || i == 1, If[l == m, 1, 0], If[i<2, 0, b[n, i - 1, l]] + b[n - i, Min[n - i, i], GCD[m, LCM[l, i]]]]; If[PrimeQ[m], 1, b[m, m, 1]]];
a /@ Range[0, 60] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

a(n) = 1 <=> n in { A000961 }. - Alois P. Heinz, Sep 26 2019

A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 5, 5, 7, 7, 10, 10, 14, 14, 17, 19, 24, 24, 32, 33, 42, 43, 58, 59, 75, 79, 98, 104, 124, 128, 156, 166, 196, 204, 239, 251, 292, 306, 352, 372, 426, 445, 514, 543, 616, 652, 745, 790, 896, 960, 1080, 1162, 1311, 1400, 1574, 1692, 1892

Offset: 0

Gus Wiseman, Jan 23 2021

nonn

The only strict partitions counted are (), (1), and (2,1).

Is there a simple generating function?

			The a(1) = 1 through a(9) = 7 partitions:
  1  11  21   22    311    2211    331      2222      333
         111  1111  2111   111111  2221     4211      4221
                    11111          4111     221111    51111
                                   211111   311111    222111
                                   1111111  11111111  321111
                                                      21111111
                                                      111111111

Note: Heinz numbers are given in parentheses below.
The reciprocal version is A143773 (A316428), with strict case A340830.
The case where length also divides n is A326842 (A326847).
The Heinz numbers of these partitions are A340606.
The version for factorizations is A340851, with reciprocal version A340853.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length/max divides n (A316413/A326836).
A067539 counts partitions with integer geometric mean (A326623).
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A330950 = partitions of n whose Heinz number is divisible by n (A324851).
Cf. A000041, A003114, A006141, A033630, A064174, A074761, A102627, A200750, A237984, A298423, A340827.

Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(Length[#]/#)&]],{n,0,30}]

A326852 Number of non-constant integer partitions of n whose length and maximum both divide n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 16, 0, 1, 7, 11, 0, 30, 0, 40, 18, 1, 0, 201, 0, 1, 38, 124, 0, 426, 0, 211, 73, 1, 48, 1391, 0, 1, 131, 1741, 0, 1774, 0, 951, 2145, 1, 0, 8345, 0, 1853, 381, 2382, 0, 6718, 2761, 10633, 623, 1, 0, 68037

Offset: 0

Gus Wiseman, Jul 26 2019

nonn

The Heinz numbers of these partitions are given by A326838.

			The a(6) = 1 through a(16) = 11 partitions (empty columns not shown):
  (321)  (4211)  (52111)  (633)     (7211111)  (53322)  (8332)
                          (642)                (53331)  (8422)
                          (651)                (54222)  (8431)
                          (4332)               (54321)  (8521)
                          (4422)               (54411)  (8611)
                          (4431)               (55221)  (42222211)
                          (6222)               (55311)  (43222111)
                          (6321)                        (43321111)
                          (6411)                        (44221111)
                          (322221)                      (44311111)
                          (332211)                      (82111111)
                          (333111)
                          (422211)
                          (432111)
                          (441111)
                          (621111)

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

The possibly constant case is A326843.
The strict case is A326851.
Cf. A018818, A047993, A067538, A326837, A326838, A326842, A326849.

Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&Divisible[n,Length[#]]&&Divisible[n,Max[#]]&]],{n,0,30}]

A340611 Number of integer partitions of n of length 2^k where k is the greatest part.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 29, 32, 34, 36, 38, 41, 42, 45, 47, 50, 52, 56, 58, 63, 66, 71, 75, 83, 88, 98, 106, 118, 128, 143, 155, 173, 188, 208, 226, 250, 270, 297, 321, 350

Offset: 0

Gus Wiseman, Jan 28 2021

nonn

Also the number of integer partitions of n with maximum 2^k where k is the length.

			The partitions for n = 12, 14, 16, 22, 24:
  32211111  32222111  32222221  33333322          33333333
  33111111  33221111  33222211  33333331          4222221111111111
            33311111  33322111  4222111111111111  4322211111111111
                      33331111  4321111111111111  4332111111111111
                                4411111111111111  4422111111111111
                                                  4431111111111111
The conjugate partitions:
  (8,2,2)  (8,3,3)  (8,4,4)  (8,7,7)     (8,8,8)
  (8,3,1)  (8,4,2)  (8,5,3)  (8,8,6)     (16,3,3,2)
           (8,5,1)  (8,6,2)  (16,2,2,2)  (16,4,2,2)
                    (8,7,1)  (16,3,2,1)  (16,4,3,1)
                             (16,4,1,1)  (16,5,2,1)
                                         (16,6,1,1)

Note: A-numbers of Heinz-number sequences are in parentheses below.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length/max divides n (A316413/A326836).
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
A340689 have a factorization of length 2^max.
A340690 have a factorization of maximum 2^length.
Cf. A003114, A006141, A064174, A098124, A117409, A200750, A340385, A340387, A340599, A340601.

Table[Length[Select[IntegerPartitions[n],Length[#]==2^Max@@#&]],{n,0,30}]

A326838 Heinz numbers of non-constant integer partitions whose length and maximum both divide their sum.

Original entry on oeis.org

30, 84, 264, 273, 286, 325, 351, 364, 390, 441, 490, 525, 624, 756, 784, 810, 840, 874, 900, 988, 1000, 1173, 1197, 1254, 1330, 1425, 1495, 1632, 1771, 2079, 2156, 2178, 2204, 2294, 2310, 2420, 2475, 2750, 2958, 3219, 3393, 3648, 3726, 3770, 3864, 3944, 4042

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 A326852.

			The sequence of terms together with their prime indices begins:
    30: {1,2,3}
    84: {1,1,2,4}
   264: {1,1,1,2,5}
   273: {2,4,6}
   286: {1,5,6}
   325: {3,3,6}
   351: {2,2,2,6}
   364: {1,1,4,6}
   390: {1,2,3,6}
   441: {2,2,4,4}
   490: {1,3,4,4}
   525: {2,3,3,4}
   624: {1,1,1,1,2,6}
   756: {1,1,2,2,2,4}
   784: {1,1,1,1,4,4}
   810: {1,2,2,2,2,3}
   840: {1,1,1,2,3,4}
   874: {1,8,9}
   900: {1,1,2,2,3,3}
   988: {1,1,6,8}

The possibly constant case is A326837.

Cf. A001222, A047993, A056239, A061395, A067538, A112798, A316413, A326836, A326843, A326847, A326848, A326851.

Select[Range[1000],With[{y=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},!SameQ@@y&&Divisible[Total[y],Max[y]]&&Divisible[Total[y],Length[y]]]&]

A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 3, 0, 4, 3, 4, 0, 8, 0, 10, 0, 11, 12, 19, 0, 0, 22, 0, 0, 46, 23, 56, 0, 64, 66, 86, 0, 125, 104, 135, 0, 196, 111, 230, 0, 0, 274, 353, 0, 0, 0, 563, 0, 687, 0, 974, 0, 1039, 1052, 1290, 0, 1473, 1511, 0, 0, 2707, 1614, 2664, 0

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. The Heinz numbers of these partitions are squarefree numbers divisible by the sum of their prime indices.

			The a(6) = 1 through a(19) = 10 partitions (empty columns indicated by dots, A = 10, B = 11):
  321  43   .  .  631   65    .  76    941   A32    .  A7     .  B8
       421        4321  542      643   6431  6432      764       865
                        5321     652   7421  9321      872       874
                                 6421        54321     971       982
                                                       7532      A81
                                                       7541      8542
                                                       7631      8632
                                                       74321     8641
                                                                 8731
                                                                 85321

Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of zeros are 2 and A013929.
The non-strict version is A330950 (A324851) q.v.
A000009 counts strict partitions.
A003963 multiplies together prime indices.
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A057568 counts partitions whose product is divisible by their sum (A326149).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A102627 counts strict partitions whose length divides sum.
A112798 lists the prime indices of each positive integer.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A324925 counts partitions whose Heinz number is divisible by their product.
A326842 counts partitions whose parts and length all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A330952 counts partitions whose Heinz number is divisible by all parts.
A340828 counts strict partitions with length divisible by maximum.
A340830 counts strict partitions with parts divisible by length.
Cf. A052335, A064173, A064174, A096401, A143773 (A316428), A168659 (A340609/A340610), A326843 (A326837).

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@Prime/@#,n]&]],{n,30}]

cp's OEIS Frontend

A340606 Numbers whose prime indices (A112798) are all divisors of the number of prime factors (A001222).

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

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A271654 a(n) = Sum_{k|n} binomial(n-1,k-1).

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A340827 Number of strict integer partitions of n into divisors of n whose length also divides n.

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A327778 Number of integer partitions of n whose LCM is a multiple of n.

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A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

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A326852 Number of non-constant integer partitions of n whose length and maximum both divide n.

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A340611 Number of integer partitions of n of length 2^k where k is the greatest part.

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