A355733 -id:A355733 - cp's OEIS frontend

A355739 Number of ways to choose a sequence of all different divisors, one of each prime index of n (with multiplicity).

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

Offset: 1

Gus Wiseman, Jul 18 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(49) = 6 ways are: (1,2), (1,4), (2,1), (2,4), (4,1), (4,2).
The a(182) = 5 ways are: (1,2,3), (1,2,6), (1,4,2), (1,4,3), (1,4,6).
The a(546) = 2 ways are: (1,2,4,3), (1,2,4,6).

Wikipedia, Cartesian product.

This is the strict version of A355731, firsts A355732.
For relatively prime instead of strict we have A355737, firsts A355738.
Positions of 0's are A355740.
A000005 counts divisors.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives GCD of prime indices, positions of 1's A289509.
Cf. A000720, A076610, A302796, A355535, A355537, A355733, A355735, A355741, A355742, A355744, A355748.

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

A355741 Number of ways to choose a sequence of prime factors, one of each prime index of n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2

Offset: 1

Gus Wiseman, Jul 18 2022

First differs from A355744 at a(169) = 4, A355744(169) = 3.

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 prime indices of 1131 are {2,6,10}, and the a(1131) = 4 choices are: {2,2,2}, {2,2,5}, {2,3,2}, {2,3,5}.

Wikipedia, Cartesian product.

Positions of 0's are A299174.
The version for all divisors is A355731, firsts A355732.
Choosing prime-power divisors gives A355742.
Positions of 1's are A355743.
Counting multisets instead of sequences gives A355744.
The weakly increasing case is A355745, all divisors A355735.
A001414 adds up distinct prime factors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A289509 lists numbers with relatively prime prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A061395, A076610, A120383, A335433, A355733, A355737, A355739, A355746, A355747.

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

Totally multiplicative with a(prime(k)) = A001221(k).

A355529 Numbers of which it is not possible to choose a different prime factor of each prime index (with multiplicity).

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106

Offset: 1

Gus Wiseman, Jul 24 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.

Includes all even numbers.

			The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}

The odd case is A355535.
The case of all divisors (not just primes) is A355740, zeros of A355739.
These choices are variously counted by A355741, A355744, A355745.
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.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A000720, A076610, A318979, A335433, A335448, A355733.

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

A355740 Numbers of which it is not possible to choose a different divisor of each prime index.

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 48, 50, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188

Offset: 1

Gus Wiseman, Jul 22 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.

By Hall's marriage theorem, k is a term if and only if there is a sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, k is divisible by a member of A370348. - Robert Israel, Feb 15 2024

			The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   48: {1,1,1,1,2}
For example, the choices of a divisor of each prime index of 90 are: (1,1,1,1), (1,1,1,3), (1,1,2,1), (1,1,2,3), (1,2,1,1), (1,2,1,3), (1,2,2,1), (1,2,2,3). But none of these has all distinct elements, so 90 is in the sequence.

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

Positions of 0's in A355739.
The case of just prime factors (not all divisors) is A355529, odd A355535.
The unordered case is counted by A355733, firsts A355734.
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.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
Cf. A000720, A076610, A335433, A335448, A340827, A355737, A355749, A370348.

filter:= proc(n) uses numtheory, GraphTheory; local B, S, F, D, E, G, t, d;
  F:= ifactors(n)[2];
  F:= map(t -> [pi(t[1]), t[2]], F);
  D:= `union`(seq(divisors(t[1]), t = F));
  F:= map(proc(t) local i; seq([t[1], i], i=1..t[2]) end proc, F);
  if nops(D) < nops(F) then return false fi;
  E:= {seq(seq({t, d}, d=divisors(t[1])), t = F)};
  S:= map(t -> convert(t, name), [op(F), op(D)]);
  E:= map(e -> map(convert, e, name), E);
  G:= Graph(S, E);
  B:= BipartiteMatching(G);
  B[1] = nops(F);
end proc:
remove(filter, [$1..200]); # Robert Israel, Feb 15 2024

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

We have A001221(a(n)) >= A303975(a(n)).

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).

Original entry on oeis.org

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)).

A239312 Number of condensed integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 9, 10, 14, 16, 23, 27, 33, 41, 51, 62, 75, 93, 111, 134, 159, 189, 226, 271, 317, 376, 445, 520, 609, 714, 832, 972, 1129, 1304, 1520, 1753, 2023, 2326, 2692, 3077, 3540, 4050, 4642, 5298, 6054, 6887, 7854, 8926, 10133, 11501, 13044

Offset: 0

Clark Kimberling, Mar 15 2014

nonn

Suppose that p is a partition of n. Let x(1), x(2), ..., x(k) be the distinct parts of p, and let m(i) be the multiplicity of x(i) in p. Let c(p) be the partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n. Call a partition q of n a condensed partition of n if q = c(p) for some partition p of n. Then a(n) is the number of distinct condensed partitions of n. Note that c(p) = p if and only if p has distinct parts and that condensed partitions can have repeated parts.

Also the number of integer partitions of n such that it is possible to choose a different divisor of each part. For example, the partition (6,4,4,1) has choices (3,2,4,1), (3,4,2,1), (6,2,4,1), (6,4,2,1) so is counted under a(15). - Gus Wiseman, Mar 12 2024

			a(5) = 3 gives the number of partitions of 5 that result from condensations as shown here: 5 -> 5, 41 -> 41, 32 -> 32, 311 -> 32, 221 -> 41, 2111 -> 32, 11111 -> 5.
From _Gus Wiseman_, Mar 12 2024: (Start)
The a(1) = 1 through a(9) = 10 condensed partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (2,2)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                   (3,1)  (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (3,2,2)  (7,1)    (8,1)
                                          (4,2,1)  (3,3,2)  (4,3,2)
                                                   (4,2,2)  (4,4,1)
                                                   (4,3,1)  (5,2,2)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..100 (first 84 terms from Manfred Scheucher)
Manfred Scheucher, Python Script

The strict case is A000009.
These partitions have ranks A368110, complement A355740.
The complement is counted by A370320.
The version for prime factors (not all divisors) is A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
For a unique choice we have A370595, ranks A370810.
For multiple choices we have A370803, ranks A370811.
The case without ones is A370805, complement A370804.
The version for factorizations is A370814, complement A370813.
A000005 counts divisors.
A000041 counts integer partitions.
A237685 counts partitions of depth 1, or A353837 if we include depth 0.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A355535, A355733, A355739, A367867, A368097, A368414, A370583, A370584, A370594, A370806, A370808.

b:= proc(n,i) option remember; `if`(n=0, {[]},
      `if`(i=1, {[n]}, {seq(map(x-> `if`(j=0, x,
       sort([x[], i*j])), b(n-i*j, i-1))[], j=0..n/i)}))
    end:
a:= n-> nops(b(n$2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 01 2019

u[n_, k_] := u[n, k] = Map[Total, Split[IntegerPartitions[n][[k]]]]; t[n_] := t[n] = DeleteDuplicates[Table[Sort[u[n, k]], {k, 1, PartitionsP[n]}]]; Table[Length[t[n]], {n, 0,   30}]
Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#],UnsameQ@@#&]]>0&]], {n,0,30}] (* Gus Wiseman, Mar 12 2024 *)

Typo in definition corrected by Manfred Scheucher, May 29 2015

Name edited by Gus Wiseman, Mar 13 2024

A355744 Number of multisets that can be obtained by choosing a prime factor of each prime index of n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2

Offset: 1

Gus Wiseman, Jul 18 2022

nonn

First differs from A355741 at a(169) = 3, A355741(169) = 4.

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(169) = 3 multisets are: {2,2}, {2,3}, {3,3}.
The a(507) = 3 multisets are: {2,2,2}, {2,2,3}, {2,3,3}.

Wikipedia, Axiom of choice.

Choosing from all divisors gives A355733, firsts A355734.
Counting sequences instead of multisets gives A355741.
Choosing weakly increasing sequences of divisors gives A355745.
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.
A324850 lists numbers divisible by the product of their prime indices.
A344606 counts alternating permutations of prime indices.
Cf. A000720, A076610, A120383, A289509, A335433, A340852, A355731, A355735, A355737, A355739, A355740.

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

A355732 Least k such that there are exactly n ways to choose a sequence of divisors, one of each element of the multiset of prime indices of k (with multiplicity).

Original entry on oeis.org

1, 3, 7, 9, 53, 21, 311, 27, 49, 159, 8161, 63, 38873, 933, 371, 81, 147, 477, 2177, 24483, 189, 2809, 343, 2799, 1113, 243, 57127, 16483, 441, 1431, 6531, 73449, 2597, 567, 96721, 8427, 1029, 8397, 3339, 15239, 729, 49449, 1323, 19663, 4293, 2401, 19593, 7791

Offset: 1

Gus Wiseman, Jul 21 2022

nonn

This is the position of first appearance of n in A355731.
Appears to be a subset of A353397.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
      1: {}
      3: {2}
      7: {4}
      9: {2,2}
     53: {16}
     21: {2,4}
    311: {64}
     27: {2,2,2}
     49: {4,4}
    159: {2,16}
   8161: {1024}
     63: {2,2,4}
For example, the choices for a(12) = 63 are:
  (1,1,1)  (1,2,2)  (2,1,4)
  (1,1,2)  (1,2,4)  (2,2,1)
  (1,1,4)  (2,1,1)  (2,2,2)
  (1,2,1)  (2,1,2)  (2,2,4)

Positions of first appearances in A355731.
Counting distinct sequences after sorting: A355734, firsts of A355733.
Requiring the result to be weakly increasing: A355736, firsts of A355735.
Requiring the result to be relatively prime: A355738, firsts of A355737.
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.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A076610, A340606, A355739, A355740, A355741, A355742, A355744, A355746, A355747, A355748.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
az=Table[Times@@Length/@Divisors/@primeMS[n],{n,1000}];
Table[Position[az,k][[1,1]],{k,mnrm[az]}]

A368110 Numbers of which it is possible to choose a different divisor of each prime index.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Gus Wiseman, Dec 15 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

By Hall's marriage theorem, k is a term if and only if there is no sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, there is no divisor of k in A370348. - Robert Israel, Feb 15 2024

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  29: {10}
  30: {1,2,3}

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

Partitions of this type are counted by A239312, complement A370320.
Positions of nonzero terms in A355739.
Complement of A355740.
For just prime divisors we have A368100, complement A355529 (odd A355535).
A000005 counts divisors.
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.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
Cf. A000720, A076610, A111774, A335433, A335448, A340852, A355733, A355734, A355737, A355749, A370348.

filter:= proc(n) uses numtheory, GraphTheory; local B,S,F,D,E,G,t,d;
  F:= ifactors(n)[2];
  F:= map(t -> [pi(t[1]),t[2]], F);
  D:= `union`(seq(divisors(t[1]), t = F));
  F:= map(proc(t) local i;seq([t[1],i],i=1..t[2]) end proc,F);
  if nops(D) < nops(F) then return false fi;
  E:= {seq(seq({t,d},d=divisors(t[1])),t = F)};
  S:= map(t -> convert(t,name), [op(F),op(D)]);
  E:= map(e -> map(convert,e,name),E);
  G:= Graph(S,E);
  B:= BipartiteMatching(G);
  B[1] = nops(F);
end proc:
select(filter, [$1..100]); # Robert Israel, Feb 15 2024

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

Heinz numbers of the partitions counted by A239312.

A381454 Number of multisets that can be obtained by choosing a strict integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 3, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 3, 3, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 3, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A357982 at a(25) = 3, A357982(25) = 4.
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.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Set multipartitions are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set multipartition {1,1,2} -> {4}.

			The a(25) = 3 multisets are: {3,3}, {1,2,3}, {1,1,2,2}.

For constant instead of strict partitions see A381453, A355733, A381455, A000688.
Positions of 1 are A003586.
The upper version is A381078, before sums A050320.
For distinct block-sums see A381634, A381633, A381806.
Multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set systems (A050326, zeros A293243) see A381441 (upper).
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
More on set systems: A050342, A116539, A296120, A318361.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A293511, A299200, A299201, A299202, A300385, A317142.

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

a(A002110(n)) = A381808(n).

cp's OEIS Frontend

A355739 Number of ways to choose a sequence of all different divisors, one of each prime index of n (with multiplicity).

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A355741 Number of ways to choose a sequence of prime factors, one of each prime index of n.

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A355529 Numbers of which it is not possible to choose a different prime factor of each prime index (with multiplicity).

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A355740 Numbers of which it is not possible to choose a different divisor of each prime index.

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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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A239312 Number of condensed integer partitions of n.

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A355744 Number of multisets that can be obtained by choosing a prime factor of each prime index of n.

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A355732 Least k such that there are exactly n ways to choose a sequence of divisors, one of each element of the multiset of prime indices of k (with multiplicity).

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