A326841 -id:A326841 - cp's OEIS frontend

A355731 Number of ways to choose a sequence of divisors, one of each element of the multiset of prime indices of n (row n of A112798).

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 2, 4, 3, 4, 1, 2, 4, 4, 2, 6, 2, 3, 2, 4, 4, 8, 3, 4, 4, 2, 1, 4, 2, 6, 4, 6, 4, 8, 2, 2, 6, 4, 2, 8, 3, 4, 2, 9, 4, 4, 4, 5, 8, 4, 3, 8, 4, 2, 4, 6, 2, 12, 1, 8, 4, 2, 2, 6, 6, 6, 4, 4, 6, 8, 4, 6, 8, 4, 2, 16, 2, 2, 6, 4, 4

Offset: 1

Gus Wiseman, Jul 16 2022

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 a(15) = 4 choices are: (1,1), (1,3), (2,1), (2,3).
The a(18) = 4 choices are: (1,1,1), (1,1,2), (1,2,1), (1,2,2).

Wikipedia, Cartesian product.

Positions of 1's are A000079.
Dominated by A003963 (cf. A049820), with equality at A003586.
Positions of first appearances are A355732.
Counting distinct sequences after sorting gives A355733, firsts A355734.
Requiring the result to be weakly increasing gives A355735, firsts A355736.
Requiring the result to be relatively prime gives A355737, firsts A355738.
Requiring the choices to be distinct gives A355739, zeros A355740.
For prime divisors A355741, prime-powers A355742, weakly increasing A355745.
Choosing divisors of each of 1..n and resorting gives A355747.
An ordered version (using standard order compositions) is A355748.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A340852 lists numbers that can be factored into divisors of bigomega.
Cf. A000720, A061395, A076610, A316524, A326841, A335433, A335448, A340606, A344616, A345957, A355744, A355746.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Length/@Divisors/@primeMS[n],{n,100}]

a(n) = Product_{k=1..A001222(n)} A000005(A112798(n,k)).

A326836 Heinz numbers of integer partitions whose maximum part divides their sum.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 70, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 135, 137, 139, 144, 149, 150, 151

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), so these are numbers whose maximum prime index divides their sum of prime indices.

The enumeration of these partitions by sum is given by A067538.

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}
   31: {11}
   32: {1,1,1,1,1}

Cf. A018818, A047993, A056239, A061395, A067538, A112798, A316413.

Cf. A326837, A326838, A326839/A326840, A326841, A326843, A326850.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],Divisible[Total[primeMS[#]],Max[primeMS[#]]]&]

A348551 Heinz numbers of integer partitions whose mean is not an integer.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 20, 24, 26, 33, 35, 36, 38, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 80, 86, 92, 93, 95, 96, 102, 104, 106, 108, 112, 114, 117, 119, 120, 122, 123, 124, 126, 130, 132, 135, 136, 140, 141, 142

Offset: 1

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms and their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  42: {1,2,4}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}

A version counting nonempty subsets is A000079 - A051293.
A version counting factorizations is A001055 - A326622.
A version counting compositions is A011782 - A271654.
A version for prime factors is A175352, complement A078175.
A version for distinct prime factors A176587, complement A078174.
The complement is A316413, counted by A067538, strict A102627.
The geometric version is the complement of A326623.
The conjugate version is the complement of A326836.
These partitions are counted by A349156.
A000041 counts partitions.
A001222 counts prime factors with multiplicity.
A018818 counts partitions into divisors, ranked by A326841.
A143773 counts partitions into multiples of the length, ranked by A316428.
A236634 counts unbalanced partitions.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement A237984.
Cf. A067539, A096199, A098743, A175397, A175761, A289508, A289509, A290103, A326028, A326645, A326837.

q:= n-> (l-> nops(l)=0 or irem(add(i, i=l), nops(l))>0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..142])[];  # Alois P. Heinz, Nov 19 2021

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

A326842 Number of integer partitions of n whose parts all divide n and whose length also divides n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 21, 2, 5, 6, 9, 2, 22, 2, 21, 6, 5, 2, 134, 3, 5, 6, 23, 2, 157, 2, 27, 6, 5, 6, 478, 2, 5, 6, 208, 2, 224, 2, 31, 63, 5, 2, 1720, 3, 30, 6, 34, 2, 322, 6, 295, 6, 5, 2, 13899, 2, 5, 68, 126, 8, 429, 2, 42, 6, 358, 2, 19959, 2

Offset: 0

Gus Wiseman, Jul 26 2019

nonn

The Heinz numbers of these partitions are given by A326847.

			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) = 21 partitions:
  (12)
  (6,6)
  (4,4,4)
  (6,3,3)
  (6,4,2)
  (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..419

Partitions using divisors are A018818.
Partitions whose length divides their sum are A067538.
Cf. A047993, A102627, A316413, A326841, A326843, A326847.

Table[Length[Select[IntegerPartitions[n,All,Divisors[n]],Divisible[n,Length[#]]&]],{n,1,30}]

A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 2, 4, 5, 6, 6, 7, 8, 11, 10, 13, 17, 18, 21, 24, 27, 30, 35, 39, 46, 53, 61, 68, 79, 87, 97, 110, 123, 139, 157, 175, 196, 222, 247, 278, 312, 347, 385, 433, 476, 531, 586, 651, 720, 800, 883, 979, 1085, 1200, 1325, 1464, 1614, 1777

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

			The a(1) = 1 through a(16) = 10 partitions (A..G = 10..16):
  1  2  3   4  5   6    7   8   9    A     B    C    D    E     F      G
        21     41  42   43  62  63   64    65   84   85   86    87     A6
                   321  61      81   82    83   A2   A3   A4    A5     C4
                                621  631   A1   642  C1   C2    C3     E2
                                     4321  632  651  643  653   E1     943
                                           641  921  652  932   654    952
                                                     931  941   942    961
                                                          8321  951    C31
                                                                C21    8431
                                                                8421   8521
                                                                54321

FindStat, St000010: The length of the partition.
FindStat, St000147: The largest part of an integer partition.
FindStat, St000784: The maximum of the length and the largest part of the integer partition.

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict version is A168659 (A340609/A340610).
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A064173 counts partitions of positive/negative rank (A340787/A340788).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A096401 counts strict partition with length equal to minimum.
A102627 counts strict partitions with length dividing sum.
A326842 counts partitions whose length and parts all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340829 counts strict partitions with Heinz number divisible by sum.
A340830 counts strict partitions with all parts divisible by length.
Cf. A003114, A006141, A064174, A117409, A143773 (A316428), A200750, A326843 (A326837), A330950 (A324851).

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

A326847 Heinz numbers of integer partitions of m >= 0 using divisors of m whose length also divides m.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197

Offset: 1

Gus Wiseman, Jul 26 2019

nonn

First differs from A071139, A089352 and A086486 in lacking 60. First differs from A326837 in lacking 268.
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 A326842.

			The sequence of terms together with their prime indices begins:
   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}
  29: {10}
  30: {1,2,3}
  31: {11}
  32: {1,1,1,1,1}
  37: {12}

R. J. Mathar, Table of n, a(n) for n = 1..489

Intersection of A326841 and A316413.

Cf. A001222, A018818, A056239, A067538, A112798, A316413, A326836, A326842.

isA326847 := proc(n)
    psigsu := A056239(n) ;
    for ifs in ifactors(n)[2] do
        p := op(1,ifs) ;
        psig := numtheory[pi](p) ;
        if modp(psigsu,psig) <> 0 then
            return false;
        end if;
    end do:
    psigle := numtheory[bigomega](n) ;
    if modp(psigsu,psigle) = 0 then
        true;
    else
        false;
    end if;
end proc:
n := 1:
for i from 2 to 3000 do
    if isA326847(i) then
        printf("%d %d\n",n,i);
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019

Select[Range[2,100],With[{y=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Divisible[Total[y],Length[y]]&&And@@IntegerQ/@(Total[y]/y)]&]

Intersection of A326841 and A316413.

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

Original entry on oeis.org

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).

A387111 Number of ways to choose a sequence of distinct positive integers, one in the initial interval of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 1, 4, 0, 2, 2, 5, 0, 6, 3, 4, 0, 7, 0, 8, 0, 6, 4, 9, 0, 6, 5, 0, 0, 10, 1, 11, 0, 8, 6, 9, 0, 12, 7, 10, 0, 13, 2, 14, 0, 2, 8, 15, 0, 12, 2, 12, 0, 16, 0, 12, 0, 14, 9, 17, 0, 18, 10, 4, 0, 15, 3, 19, 0, 16, 4, 20, 0, 21, 11, 4, 0, 16, 4, 22

Offset: 1

Gus Wiseman, Aug 18 2025

nonn

The initial interval of a nonnegative integer x is the set {1,...,x}.
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 axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 75 are (2,3,3), with initial intervals ({1,2},{1,2,3},{1,2,3}), with choices (1,2,3), (1,3,2), (2,1,3), (2,3,1), so a(75) = 4.

Allowing repeated partitions gives A003963.
For constant instead of distinct we have A055396.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors we have A355739, zeros A355740, strict case of A355731.
For prime factors we have A355741, prime powers A355742, weakly increasing A355745.
For integer partitions we have A387110.
Positions of nonzero terms are A387112 (choosable).
Positions of 0 are A387134 (non-choosable).
A001414 adds up distinct prime divisors, counted by A001221.
A061395 gives greatest prime index.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A326841, A335433, A335448, A355733, A355737, A355747, A357980, A383706, A387120, A387133, A387135.

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

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

cp's OEIS Frontend

A355731 Number of ways to choose a sequence of divisors, one of each element of the multiset of prime indices of n (row n of A112798).

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A326836 Heinz numbers of integer partitions whose maximum part divides their sum.

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A348551 Heinz numbers of integer partitions whose mean is not an integer.

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A326842 Number of integer partitions of n whose parts all divide n and whose length also divides n.

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A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

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A326847 Heinz numbers of integer partitions of m >= 0 using divisors of m whose length also divides m.

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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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