A050336 -id:A050336 - cp's OEIS frontend

A321514 Number of ways to choose a factorization of each integer from 2 to n into factors > 1.

1, 1, 1, 2, 2, 4, 4, 12, 24, 48, 48, 192, 192, 384, 768, 3840, 3840, 15360, 15360, 61440, 122880, 245760, 245760, 1720320, 3440640, 6881280, 20643840, 82575360, 82575360, 412876800, 412876800, 2890137600, 5780275200, 11560550400, 23121100800, 208089907200

Offset: 1

Gus Wiseman, Nov 11 2018

nonn

			The a(8) = 12 ways to choose a factorization of each integer from 2 to 8:
  (2)*(3)*(4)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*2*2)

Dominates A321468.

Cf. A001055, A050336, A066723, A076716, A157612, A281113, A300383, A317144, A317145, A321467, A321470, A321471, A321472.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Array[Length[facs[#]]&,n,1,Times],{n,30}]

a(n) = Product_{k = 1..n} A001055(k).

A050340 Number of ways of factoring n with 3 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 5, 1, 5, 1, 15, 5, 5, 1, 25, 1, 5, 5, 55, 1, 25, 1, 25, 5, 5, 1, 105, 5, 5, 15, 25, 1, 35, 1, 170, 5, 5, 5, 145, 1, 5, 5, 105, 1, 35, 1, 25, 25, 5, 1, 425, 5, 25, 5, 25, 1, 105, 5, 105, 5, 5, 1, 205, 1, 5, 25, 571, 5, 35, 1, 25, 5, 35, 1, 660, 1, 5, 25, 25, 5, 35, 1, 425, 55, 5

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on the 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).

			4 = (((4))) = (((2*2))) = (((2)*(2))) = (((2))*((2))) = (((2)))*(((2))).

R. J. Mathar, Table of n, a(n) for n = 1..1295

Cf. A001055, A050336-A050341. a(p^k) = A007714. a(A002110) = A000357.

Dirichlet g.f.: Product{n=2..infinity} (1/(1-1/n^s)^A050338(n)).

a(n) = A050341(A101296(n)). - R. J. Mathar, May 26 2017

A322436 Number of pairs of factorizations of n into factors > 1 where no factor of the second properly divides any factor of the first.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 8, 1, 3, 3, 11, 1, 8, 1, 8, 3, 3, 1, 18, 3, 3, 5, 8, 1, 12, 1, 15, 3, 3, 3, 31, 1, 3, 3, 18, 1, 12, 1, 8, 8, 3, 1, 39, 3, 8, 3, 8, 1, 18, 3, 18, 3, 3, 1, 42, 1, 3, 8, 33, 3, 12, 1, 8, 3, 12, 1, 67, 1, 3, 8, 8, 3, 12, 1, 39, 11

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

			The a(12) = 8 pairs of factorizations:
  (2*2*3)|(2*2*3)
  (2*2*3)|(2*6)
  (2*2*3)|(3*4)
  (2*2*3)|(12)
    (2*6)|(12)
    (3*4)|(3*4)
    (3*4)|(12)
     (12)|(12)

Cf. A000688, A001055, A050336, A265947, A281113, A294068, A303386, A305149, A305150, A305193, A317144, A322435, A322439, A322440, A322441, A322442.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
divpropQ[x_,y_]:=And[x!=y,Divisible[x,y]];
Table[Length[Select[Tuples[facs[n],2],!Or@@divpropQ@@@Tuples[#]&]],{n,100}]

A320887 Number of multiset partitions of factorizations of n into factors > 1 such that all the parts have the same product.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 9, 1, 4, 1, 4, 2, 2, 1, 7, 3, 2, 4, 4, 1, 5, 1, 8, 2, 2, 2, 12, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 3, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 22, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 9, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 12, 1, 5, 1, 7, 5

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

			The a(36) = 12 multiset partitions:
  (2*2*3*3)    (6)*(2*3)  (6)*(6)  (36)
  (2*3)*(2*3)  (2*2*9)    (2*18)
               (2*3*6)    (3*12)
               (3*3*4)    (4*9)
                          (6*6)

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A001970, A046523, A050336, A052409, A279375, A294786, A295923, A320886, A320888, A320889.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[With[{g=GCD@@FactorInteger[n][[All,2]]},Sum[Binomial[Length[facs[n^(1/d)]]+d-1,d],{d,Divisors[g]}]],{n,100}]

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A320887(n) = if(1==n,n,my(r); sumdiv(A052409(n), d, binomial(A001055(sqrtnint(n,d)) + d - 1, d))); \\ Antti Karttunen, Nov 17 2019

a(n) = Sum_{d|A052409(n)} binomial(A001055(n^(1/d)) + d - 1, d).

a(n) = a(A046523(n)). - Antti Karttunen, Nov 17 2019

Data section extended up to term a(105) by Antti Karttunen, Nov 17 2019

A301598 Number of thrice-factorizations of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 10, 4, 4, 1, 16, 1, 4, 4, 34, 1, 16, 1, 16, 4, 4, 1, 54, 4, 4, 10, 16, 1, 22, 1, 80, 4, 4, 4, 78, 1, 4, 4, 54, 1, 22, 1, 16, 16, 4, 1, 181, 4, 16, 4, 16, 1, 54, 4, 54, 4, 4, 1, 102, 1, 4, 16, 254, 4, 22, 1, 16, 4, 22, 1, 272, 1, 4, 16, 16

Offset: 1

Gus Wiseman, Mar 24 2018

nonn

A thrice-factorization of n is a choice of a twice-factorization of each factor in a factorization of n. Thrice-factorizations correspond to intervals in the lattice form of the multiorder of integer factorizations.

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

Gus Wiseman, The (2*2*3*3) component of the lattice form of the multiorder of integer factorizations.
Gus Wiseman, The (2*2*3*3) component, version 2.

Cf. A001055, A007716, A050336, A050338, A063834, A162247, A269134, A281113, A281116, A301595, A301598, A301706.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
twifacs[n_]:=Join@@Table[Tuples[facs/@f],{f,facs[n]}];
thrifacs[n_]:=Join@@Table[Tuples[twifacs/@f],{f,facs[n]}];
Table[Length[thrifacs[n]],{n,15}]

Dirichlet g.f.: Product_{n > 1} 1/(1 - A281113(n)/n^s).

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

A320888 Number of set multipartitions (multisets of sets) of factorizations of n into factors > 1 such that all the parts have the same product.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 5, 1, 4, 2, 2, 2, 8, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 9, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 4, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

			The a(144) = 20 set multipartitions:
  (2*3*4*6)    (2*8*9)     (2*72)     (144)
  (2*6)*(2*6)  (3*6*8)     (3*48)
  (2*6)*(3*4)  (2*3*24)    (4*36)
  (3*4)*(3*4)  (2*4*18)    (6*24)
               (2*6*12)    (8*18)
               (3*4*12)    (9*16)
               (12)*(2*6)  (12)*(12)
               (12)*(3*4)

Cf. A001055, A001970, A045778, A050336, A052409, A089259, A294786, A296132, A319269, A320886, A320887, A320889.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[With[{g=GCD@@FactorInteger[n][[All,2]]},Sum[Binomial[Length[strfacs[n^(1/d)]]+d-1,d],{d,Divisors[g]}]],{n,100}]

a(n) = Sum_{d|A052409(n)} binomial(A045778(n^(1/d)) + d - 1, d).

A320889 Number of set partitions of strict factorizations of n into factors > 1 such that all the blocks have the same product.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 6, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 5, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 2, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

			The a(144) = 17 set partitions:
  (2*3*4*6)    (2*8*9)     (2*72)  (144)
  (2*6)*(3*4)  (3*6*8)     (3*48)
               (2*3*24)    (4*36)
               (2*4*18)    (6*24)
               (2*6*12)    (8*18)
               (3*4*12)    (9*16)
               (2*6)*(12)
               (3*4)*(12)

Cf. A000110, A001055, A001970, A045778, A050336, A279375, A294786, A294787, A296132, A320886, A320887, A320888.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[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@@Table[Select[sps[fac],SameQ@@Times@@@#&],{fac,strfacs[n]}]],{n,100}]

A322453 Number of factorizations of n into factors > 1 using only primes and perfect powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 5, 1, 1, 1, 3, 1

Offset: 1

Gus Wiseman, Dec 09 2018

nonn

First differs from A000688 at a(36) = 5, A000688(36) = 4.

Terms in this sequence only depend on the prime signature of n. - David A. Corneth, Dec 26 2018

			The a(144) = 13 factorizations:
  (144),
  (4*36), (9*16),
  (2*2*36), (2*8*9), (3*3*16), (4*4*9),
  (2*2*4*9), (2*3*3*8), (3*3*4*4),
  (2*2*2*2*9), (2*2*3*3*4),
  (2*2*2*2*3*3).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A000688, A000961, A001055, A001597, A025487, A050336, A284696, A294068, A320322, A322452.

perpowQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1;
pfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[pfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],Or[PrimeQ[#],perpowQ[#]]&]}]];
Table[Length[pfacs[n]],{n,100}]

A322453(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(ispower(d)||isprime(d)), s += A322453(n/d, d))); (s)); \\ Antti Karttunen, Dec 26 2018

More terms from Antti Karttunen, Dec 24 2018

A323719 Array read by antidiagonals upwards where A(n, k) is the number of orderless factorizations of n with k - 1 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 3, 1, 4, 1, 6, 1, 1, 1, 1, 2, 6, 1, 5, 1, 7, 1, 1, 1, 1, 2, 3, 10, 1, 6, 1, 8, 1, 1, 1, 1, 1, 3, 4, 15, 1, 7, 1, 9, 1, 1, 1, 1, 4, 1, 4, 5, 21, 1, 8, 1, 10, 1, 1, 1

Offset: 1

Gus Wiseman, Jan 25 2019

An orderless factorization of n with k > 1 levels of parentheses is any multiset partition of an orderless factorization of n with k - 1 levels of parentheses. If k = 1 it is just an orderless factorization of n into factors > 1.

			Array begins:
       k=0  k=1  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9  k=10 k=11 k=12
   n=1: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=2: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=3: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=4: 1    2    3    4    5    6    7    8    9   10   11   12   13
   n=5: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=6: 1    2    3    4    5    6    7    8    9   10   11   12   13
   n=7: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=8: 1    3    6   10   15   21   28   36   45   55   66   78   91
   n=9: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=10: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=11: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=12: 1    4    9   16   25   36   49   64   81  100  121  144  169
  n=13: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=14: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=15: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=16: 1    5   14   30   55   91  140  204  285  385  506  650  819
  n=17: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=18: 1    4    9   16   25   36   49   64   81  100  121  144  169
The A(12,3) = 16 orderless factorizations of 12 with 2 levels of parentheses:
  ((2*2*3))          ((2*6))      ((3*4))      ((12))
  ((2)*(2*3))        ((2)*(6))    ((3)*(4))
  ((3)*(2*2))        ((2))*((6))  ((3))*((4))
  ((2))*((2*3))
  ((2)*(2)*(3))
  ((3))*((2*2))
  ((2))*((2)*(3))
  ((3))*((2)*(2))
  ((2))*((2))*((3))

Columns: A001055 (k=1), A050336 (k=2), A050338 (k=3), A050340 (k=4).
Rows: A000027, A000217, A000290, etc.
Cf. A096751, A141268, A144150, A213427, A255906, A281113, A290353, A292504, A317145, A318564, A318565, A318812, A323718.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
lev[n_,k_]:=If[k==0,{n},Join@@Table[Union[Sort/@Tuples[lev[#,k-1]&/@fac]],{fac,facs[n]}]];
Table[Length[lev[sum-k,k]],{sum,12},{k,0,sum-1}]

cp's OEIS Frontend

A321514 Number of ways to choose a factorization of each integer from 2 to n into factors > 1.

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A050340 Number of ways of factoring n with 3 levels of parentheses.

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A322436 Number of pairs of factorizations of n into factors > 1 where no factor of the second properly divides any factor of the first.

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A320887 Number of multiset partitions of factorizations of n into factors > 1 such that all the parts have the same product.

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A301598 Number of thrice-factorizations of n.

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

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A320888 Number of set multipartitions (multisets of sets) of factorizations of n into factors > 1 such that all the parts have the same product.

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A320889 Number of set partitions of strict factorizations of n into factors > 1 such that all the blocks have the same product.

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Mathematica

A322453 Number of factorizations of n into factors > 1 using only primes and perfect powers.

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