A294788 -id:A294788 - cp's OEIS frontend

A296119 Number of ways to choose a strict factorization of each factor in a factorization of n.

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 16, 2, 3, 4, 7, 1, 12, 1, 12, 3, 3, 3, 21, 1, 3, 3, 16, 1, 12, 1, 7, 7, 3, 1, 33, 2, 7, 3, 7, 1, 16, 3, 16, 3, 3, 1, 34, 1, 3, 7, 23, 3, 12, 1, 7, 3, 12, 1, 50, 1, 3, 7, 7, 3, 12, 1, 33, 7, 3

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

			The a(24) = 16 twice-factorizations:
(2)*(2)*(2)*(3),
(2)*(2)*(2*3), (2)*(2)*(6), (2)*(3)*(4),
(2)*(2*6), (2)*(3*4), (2)*(12), (3)*(2*4), (3)*(8), (4)*(2*3), (4)*(6),
(2*3*4), (2*12), (3*8), (4*6), (24).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000009, A005117, A045778, A270995, A281113, A294788, A296118, A296120, A296121.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@(Length[Select[facs[#],UnsameQ@@#&]]&/@fac),{fac,facs[n]}],{n,100}]

A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(dA045778(n/d, d-1))));
A296119(n, m=n) = if(1==n, 1, sumdiv(n, d, if((d>1)&&(d<=m), A045778(d)*A296119(n/d, d)))); \\ Antti Karttunen, Oct 08 2018

Dirichlet g.f.: 1/Product_{n > 1}(1 - A045778(n)/n^s).

A302505 Numbers whose prime indices are squarefree and have disjoint prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 15, 16, 17, 20, 22, 24, 26, 29, 30, 31, 32, 33, 34, 40, 41, 43, 44, 47, 48, 51, 52, 55, 58, 59, 60, 62, 64, 66, 67, 68, 73, 79, 80, 82, 83, 85, 86, 88, 93, 94, 96, 101, 102, 104, 109, 110, 113, 116, 118, 120, 123, 124, 127

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
08: {{},{},{}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
13: {{1,2}}
15: {{1},{2}}
16: {{},{},{},{}}
17: {{4}}
20: {{},{},{2}}
22: {{},{3}}
24: {{},{},{},{1}}
26: {{},{1,2}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
32: {{},{},{},{},{}}

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279790, A281113, A294788, A301750, A302242, A302243.

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

A296120 Number of ways to choose a strict factorization of each factor in a strict factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 14, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 11, 3, 12, 1, 6, 3, 12, 1, 36, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

			The a(36) = 14 twice-factorizations:
(36), (4*9), (3*12), (2*18), (2*3*6),
(4)*(9), (3)*(12), (3)*(3*4), (3)*(2*6), (2)*(18), (2)*(3*6), (2)*(2*9),
(2)*(3)*(6), (2)*(3)*(2*3).

Cf. A000009, A005117, A045778, A050345, A279785, A281113, A294788, A296118, A296119, A296121.

sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@Length/@sfs/@fac,{fac,sfs[n]}],{n,100}]

Dirichlet g.f.: Product_{n > 1}(1 + A045778(n)/n^s).

A050345 Number of ways to factor n into distinct factors with one level of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 15, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 12, 3, 12, 1, 6, 3, 12, 1, 37, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3, 1, 31, 3, 3, 3, 13, 1, 31, 3, 6, 3, 3

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

First differs from A296120 at a(36) = 15, A296120(36) = 14. - Gus Wiseman, Apr 27 2025
Each "part" in parentheses is distinct from all others at the same level. Thus (3*2)*(2) is allowed but (3)*(2*2) and (3*2*2) are not.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			12 = (12) = (6*2) = (6)*(2) = (4*3) = (4)*(3) = (3*2)*(2).
From _Gus Wiseman_, Apr 26 2025: (Start)
This is the number of ways to partition a factorization of n (counted by A001055) into a set of sets. For example, the a(12) = 6 choices are:
  {{2},{2,3}}
  {{2},{6}}
  {{3},{4}}
  {{2,6}}
  {{3,4}}
  {{12}}
(End)

R. J. Mathar, Table of n, a(n) for n = 1..2519

Cf. A050346-A050350. a(A002110)=A000258.
For multisets of multisets we have A050336.
For integer partitions we have a(p^k) = A050342(k), see A001970, A089259, A261049.
For normal multiset partitions see A116539, A292432, A292444, A381996, A382214, A382216.
The case of a unique choice (positions of 1) is A166684.
Twice-partitions of this type are counted by A358914, see A270995, A281113, A294788.
For sets of multisets we have A383310 (distinct products A296118).
For multisets of sets we have we have A383311, see A296119.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A302494 gives MM-numbers of sets of sets.
A382077 counts partitions that can be partitioned into a sets of sets, ranks A382200.
A382078 counts partitions that cannot be partitioned into a sets of sets, ranks A293243.
Cf. A000009, A002865, A005117, A045782, A279785, A293511, A296120, A316439, A381441, A382201.

facs[n_]:=If[n<=1,{{}}, Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d, Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}] := Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort /@ (#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[y], UnsameQ@@#&&And@@UnsameQ@@@#&]], {y,facs[n]}],{n,30}] (* Gus Wiseman, Apr 26 2025 *)

Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A045778(n).
a(n) = A050346(A025487^(-1)(A046523(n))), where A025487^(-1) is the inverse with A025487^(-1)(A025487(n))=n. - R. J. Mathar, May 25 2017
a(n) = A050346(A101296(n)). - Antti Karttunen, May 25 2017

A294786 Number of ways to choose a set partition of a factorization of n into distinct factors greater than one such that different blocks have different products.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 12, 1, 3, 3, 5, 1, 12, 1, 5, 3, 3, 3, 11, 1, 3, 3, 12, 1, 12, 1, 5, 5, 3, 1, 19, 1, 5, 3, 5, 1, 12, 3, 12, 3, 3, 1, 26, 1, 3, 5, 9, 3, 12, 1, 5, 3, 12, 1, 26, 1, 3, 5, 5, 3, 12, 1, 19, 3, 3

Offset: 1

Gus Wiseman, Nov 08 2017

nonn

			The a(36)=11 ways are:
(2)*(3)*(6),
(2)*(3*6), (2*6)*(3), (2)*(18), (3)*(12), (4)*(9),
(2*3*6), (2*18), (3*12), (4*9), (36).

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Cf. A000258, A001055, A045778, A279375, A281113, A294787, A294788.

sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Join@@Function[fac,Select[sps[fac],UnsameQ@@Times@@@#&]]/@sfs[n]],{n,100}]

a(product of n distinct primes) = A000258(n).

a(prime^n) = A279375(n).

A296118 Number of ways to choose a factorization of each factor in a strict factorization of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 18, 3, 3, 3, 23, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 45, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 34, 3, 12, 1, 8, 3, 12, 1, 66, 1, 3, 8, 8, 3, 12, 1, 45, 8, 3

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

			The a(12) = 8 twice-factorizations are (2)*(2*3), (2)*(6), (3)*(2*2), (3)*(4), (2*2*3), (2*6), (3*4), (12).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000009, A005117, A045778, A261049, A271619, A281113, A294788, A295923, A296119, A296121.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@(Length[facs[#]]&/@f),{f,Select[facs[n],UnsameQ@@#&]}],{n,100}]

A001055(n, m=n) = if(1==n, 1, sumdiv(n, d, if((d>1)&&(d<=m), A001055(n/d, d))));
A296118(n, m=n) = ((n<=m)*A001055(n) + sumdiv(n, d, if((d>1)&&(d<=m)&&(dA001055(d)*A296118(n/d, d-1)))); \\ Antti Karttunen, Oct 08 2018

Dirichlet g.f.: Product_{n > 1}(1 + A001055(n)/n^s).

A302521 Odd numbers whose prime indices are squarefree and have disjoint prime indices. Numbers n such that the n-th multiset multisystem is a set partition.

Original entry on oeis.org

1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 41, 43, 47, 51, 55, 59, 67, 73, 79, 83, 85, 93, 101, 109, 113, 123, 127, 137, 139, 141, 143, 145, 149, 155, 157, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 215, 219, 221, 233, 241, 249, 255, 257, 269, 271

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set partitions.
01: {}
03: {{1}}
05: {{2}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
51: {{1},{4}}
55: {{2},{3}}
59: {{7}}
67: {{8}}
73: {{2,4}}
79: {{1,5}}
83: {{9}}
85: {{2},{4}}
93: {{1},{5}}

Cf. A000110, A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279790, A281113, A294788, A301750, A302242, A302243, A302505.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],UnsameQ@@Join@@primeMS/@primeMS[#]&]

A294787 Number of ways to choose a set partition of a factorization of n into distinct factors greater than one.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 12, 1, 3, 3, 5, 1, 12, 1, 5, 3, 3, 3, 12, 1, 3, 3, 12, 1, 12, 1, 5, 5, 3, 1, 19, 1, 5, 3, 5, 1, 12, 3, 12, 3, 3, 1, 26, 1, 3, 5, 10, 3, 12, 1, 5, 3, 12, 1, 26, 1, 3, 5, 5, 3, 12, 1, 19, 3, 3

Offset: 1

Gus Wiseman, Nov 08 2017

nonn

Cf. A001055, A045778, A279375, A281113, A294617, A294786, A294788.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Total[BellB/@Length/@strfacs[n]],{n,100}]

A294786(n) <= a(n) <= A294788(n).

A301750 Number of rooted twice-partitions of n where the composite rooted partition is strict.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 18, 29, 42, 61, 86, 127, 181, 257, 352, 489, 668, 935, 1270, 1730, 2312, 3101, 4112, 5533, 7345, 9742, 12785, 16793, 21821, 28452, 36908, 48108, 62198, 80337, 103081, 132372, 168805, 215247, 273678

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n.

			The a(8) = 18 rooted twice-partitions where the composite rooted partition is strict:
(6), (51), (42), (321),
(5)(), (41)(), (32)(), (4)(1), (3)(2),
(4)()(), (31)()(), (3)(1)(),
(3)()()(), (21)()()(), (2)(1)()(),
(2)()()()(),
(1)()()()()(),
()()()()()()().

Cf. A002865, A063834, A093637, A127524, A279790, A294788, A301422, A301462, A301467, A301480, A301706.

twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],UnsameQ@@Join@@#&]//Length,{n,30}]

A320886 Number of multiset partitions of integer partitions of n where all parts have the same product.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 25, 33, 54, 73, 107, 140, 207, 264, 369, 479, 652, 828, 1112, 1400, 1848, 2326, 3009, 3762, 4856, 6020, 7648, 9478, 11942, 14705, 18427, 22576, 28083, 34350, 42429, 51714, 63680, 77289, 94618, 114648, 139773, 168799, 205144, 247128, 299310, 359958, 434443, 521255, 627812, 751665, 902862

Offset: 0

Gus Wiseman, Oct 23 2018

nonn

			The a(1) = 1 through a(6) = 25 multiset partitions:
  (1)  (2)     (3)        (4)           (5)              (6)
       (11)    (12)       (13)          (14)             (15)
       (1)(1)  (111)      (22)          (23)             (24)
               (1)(11)    (112)         (113)            (33)
               (1)(1)(1)  (1111)        (122)            (114)
                          (2)(2)        (1112)           (123)
                          (1)(111)      (11111)          (222)
                          (11)(11)      (2)(12)          (1113)
                          (1)(1)(11)    (1)(1111)        (1122)
                          (1)(1)(1)(1)  (11)(111)        (3)(3)
                                        (1)(1)(111)      (11112)
                                        (1)(11)(11)      (111111)
                                        (1)(1)(1)(11)    (12)(12)
                                        (1)(1)(1)(1)(1)  (2)(112)
                                                         (2)(2)(2)
                                                         (1)(11111)
                                                         (11)(1111)
                                                         (111)(111)
                                                         (1)(1)(1111)
                                                         (1)(11)(111)
                                                         (11)(11)(11)
                                                         (1)(1)(1)(111)
                                                         (1)(1)(11)(11)
                                                         (1)(1)(1)(1)(11)
                                                         (1)(1)(1)(1)(1)(1)

Cf. A001055, A001970, A045778, A050336, A279375, A294617, A294786, A294787, A294788, A320887, A320888, A320889.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],SameQ@@Times@@@#&]],{n,8}]

G(n)={my(M=Map()); for(k=1, n, forpart(p=k, my(t=vecprod(Vec(p)), z); mapput(M, t, if(mapisdefined(M, t, &z), z, 0) + x^k))); M}
a(n)=if(n==0, 1, vecsum(apply(p->EulerT(Vecrev(p/x, n))[n], Mat(G(n))[,2]))) \\ Andrew Howroyd, Oct 26 2018

a(13)-a(50) from Andrew Howroyd, Oct 26 2018

cp's OEIS Frontend

A296119 Number of ways to choose a strict factorization of each factor in a factorization of n.

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A302505 Numbers whose prime indices are squarefree and have disjoint prime indices.

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A296120 Number of ways to choose a strict factorization of each factor in a strict factorization of n.

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A050345 Number of ways to factor n into distinct factors with one level of parentheses.

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A294786 Number of ways to choose a set partition of a factorization of n into distinct factors greater than one such that different blocks have different products.

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A296118 Number of ways to choose a factorization of each factor in a strict factorization of n.

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A302521 Odd numbers whose prime indices are squarefree and have disjoint prime indices. Numbers n such that the n-th multiset multisystem is a set partition.

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A294787 Number of ways to choose a set partition of a factorization of n into distinct factors greater than one.

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A301750 Number of rooted twice-partitions of n where the composite rooted partition is strict.