A347456 -id:A347456 - cp's OEIS frontend

A348381 Number of inseparable factorizations of n that are not a twin (x*x).

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

First differs from A347706 at a(216) = 3, A347706(216) = 4.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is inseparable if it has no permutation that is an anti-run, meaning there are always adjacent equal parts. Alternatively, a multiset is inseparable if its maximal multiplicity is at most one plus the sum of its remaining multiplicities.

			The a(n) factorizations for n = 96, 192, 384, 576:
  2*2*2*12      3*4*4*4         4*4*4*6           4*4*4*9
  2*2*2*2*6     2*2*2*24        2*2*2*48          2*2*2*72
  2*2*2*2*2*3   2*2*2*2*12      2*2*2*2*24        2*2*2*2*36
                2*2*2*2*2*6     2*2*2*2*3*8       2*2*2*2*4*9
                2*2*2*2*3*4     2*2*2*2*4*6       2*2*2*2*6*6
                2*2*2*2*2*2*3   2*2*2*2*2*12      2*2*2*2*2*18
                                2*2*2*2*2*2*6     2*2*2*2*3*12
                                2*2*2*2*2*3*4     2*2*2*2*2*2*9
                                2*2*2*2*2*2*2*3   2*2*2*2*2*3*6
                                                  2*2*2*2*2*2*3*3

Positions of nonzero terms are A046099.
Partitions not of this type are counted by A325534 - A000035.
Partitions of this type are counted by A325535 - A000035.
Allowing twins gives A333487.
The case without an alternating permutation is A347706, with twins A348380.
The complement is counted by A348383, without twins A335434.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations of sets.
A008480 counts permutations of prime indices, strict A335489.
A025047 counts alternating or wiggly compositions.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A344654 counts non-twin partitions without an alternating permutation.
A348382 counts non-anti-run compositions that are not a twin.
A348611 counts anti-run ordered factorizations.
Cf. A038548, A336102, A344614, A344653, A344740, A347050, A347437, A347438, A347456, A348379, A348609.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],!MatchQ[#,{x_,x_}]&&Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]=={}&]],{n,100}]

a(n > 1) = A333487(n) - A010052(n).

a(2^n) = A325535(n) - 1 for odd n, otherwise A325535(n).

A348383 Number of factorizations of n that are either separable (have an anti-run permutation) or are a twin (x*x).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

First differs from A347050 at a(216) = 28, A347050(216) = 27.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.

			The a(216) = 28 factorizations:
  (2*2*2*3*3*3)  (2*2*2*3*9)  (2*2*6*9)   (3*8*9)   (3*72)   (216)
                 (2*2*3*3*6)  (2*3*4*9)   (4*6*9)   (4*54)
                 (2*3*3*3*4)  (2*3*6*6)   (2*2*54)  (6*36)
                              (3*3*4*6)   (2*3*36)  (8*27)
                              (2*2*3*18)  (2*4*27)  (9*24)
                              (2*3*3*12)  (2*6*18)  (12*18)
                                          (2*9*12)  (2*108)
                                          (3*3*24)
                                          (3*4*18)
                                          (3*6*12)
The a(270) = 20 factorizations:
  (2*3*3*3*5)  (2*3*5*9)   (5*6*9)   (3*90)   (270)
               (3*3*5*6)   (2*3*45)  (5*54)
               (2*3*3*15)  (2*5*27)  (6*45)
                           (2*9*15)  (9*30)
                           (3*3*30)  (10*27)
                           (3*5*18)  (15*18)
                           (3*6*15)  (2*135)
                           (3*9*10)

Positions of 1's are 1 and A000040.
Not requiring separability gives A010052 for n > 1.
Positions of 2's are A323644.
Partitions of this type are counted by A325534(n) + A000035(n + 1).
Partitions of this type are ranked by A335433 \/ A001248.
Partitions not of this type are counted by A325535(n) - A000035(n + 1).
Partitions not of this type are ranked by A345193 = A335448 \ A001248.
Not allowing twins gives A335434, complement A333487,
The case with an alternating permutation is A347050, no twins A348379.
The case without an alternating permutation is A347706, no twins A348380.
The complement is counted by A348381.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A003242 counts anti-run compositions, ranked by A333489.
A025047 counts alternating or wiggly compositions.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A119620, A138364, A336103, A344654, A344740, A347437, A347438, A347456, A347458, A348382.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sepQ[m_]:=Select[Permutations[m],!MatchQ[#,{_,x_,x_,_}]&]!={};
Table[Length[Select[facs[n],MatchQ[#,{x_,x_}]||sepQ[#]&]],{n,100}]

a(n > 1) = A335434(n) + A010052(n), where A010052(n) = 1 if n is a perfect square, otherwise 0.

A347447 Number of strict factorizations of n with alternating product > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 4, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 23 2021

nonn

A strict factorization of n is an increasing sequence of distinct positive integers > 1 with product n.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
All such factorizations must have odd length.

			The a(720) = 30 factorizations:
  (2*4*90)     (3*4*60)   (4*5*36)   (5*6*24)  (6*8*15)   (8*9*10)  (720)
  (2*5*72)     (3*5*48)   (4*6*30)   (5*8*18)  (6*10*12)
  (2*6*60)     (3*6*40)   (4*9*20)   (5*9*16)
  (2*8*45)     (3*8*30)   (4*10*18)
  (2*9*40)     (3*10*24)  (4*12*15)
  (2*10*36)    (3*12*20)
  (2*12*30)    (3*15*16)
  (2*15*24)
  (2*18*20)
  (2*3*120)
  (2*3*4*5*6)

Allowing any alternating product gives A045778.
The reverse additive version (or restriction to powers of 2) is A067659.
The non-strict version is A339890.
Allowing equal parts and any alternating product < 1 gives A347440.
Allowing equal parts and any alternating product >= 1 gives A347456.
A046099 counts factorizations with no alternating permutations.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A339846 counts even-length factorizations.
A347437 counts factorizations with integer alternating product.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A000009, A005117, A030059, A119620, A119899, A330972, A344608, A347438, A347439, A347442, A347463.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],UnsameQ@@#&&altprod[#]>1&]],{n,100}]

A347708 Number of distinct possible alternating products of odd-length factorizations of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 11 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
Note that it is sufficient to look at only length-1 and length-3 factorizations; cf. A347709.

			Representative factorizations for each of the a(180) = 7 alternating products:
  (2*2*3*3*5) -> 5
     (2*2*45) -> 45
     (2*3*30) -> 20
     (2*5*18) -> 36/5
     (2*9*10) -> 20/9
     (3*4*15) -> 45/4
        (180) -> 180

Antti Karttunen, Table of n, a(n) for n = 1..65537

The version for partitions is A028310, reverse A347707.
Positions of 1's appear to be A037143 \ {1}.
The even-length version for n > 1 is A072670, strict A211159.
Counting only integers appears to give A293234, with evens A046951.
This is the odd-length case of A347460, reverse A038548.
The any-length version for partitions is A347461, reverse A347462.
The length-3 case is A347709.
A001055 counts factorizations (strict A045778, ordered A074206).
A056239 adds up prime indices, row sums of A112798.
A276024 counts distinct positive subset-sums of partitions.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions.
A347050 = factorizations w/ an alternating permutation, complement A347706.
A347441 counts odd-length factorizations with integer alternating product.
Cf. A002033, A103919, A108917, A119620, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],OddQ[Length[#]]&]]],{n,100}]

altprod(facs) = prod(i=1,#facs,facs[i]^((-1)^(i-1)));
A347708aux(n, m=n, facs=List([])) = if(1==n, if((#facs)%2, altprod(facs), 0), my(newfacs, r, rats=List([])); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); r = A347708aux(n/d, d, newfacs); if(r, rats = concat(rats,r)))); (rats));
A347708(n) = if(1==n,0,#Set(A347708aux(n))); \\ Antti Karttunen, Jan 29 2025

Conjecture: For n > 1, a(n) = 1 + A347460(n) - A038548(n) + A072670(n).

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A347048 Number of even-length ordered factorizations of n with integer alternating product.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 4, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 6, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 11, 0, 0, 0, 1, 0, 0, 0, 11, 0, 0, 1, 1, 0, 0, 0, 6, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 8, 0, 1, 1, 7, 0, 0, 0, 1, 0

Offset: 1

Gus Wiseman, Oct 10 2021

nonn

An ordered factorization of n is a sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(n) ordered factorizations for n = 16, 32, 36, 48, 64, 96:
  4*4       8*4       6*6       12*4      8*8           24*4
  8*2       16*2      12*3      24*2      16*4          48*2
  2*2*2*2   2*2*4*2   18*2      2*2*6*2   32*2          3*2*8*2
            4*2*2*2   2*2*3*3   3*2*4*2   2*2*4*4       4*2*6*2
                      2*3*3*2   4*2*3*2   2*2*8*2       6*2*4*2
                      3*2*2*3   6*2*2*2   2*4*4*2       8*2*3*2
                      3*3*2*2             4*2*2*4       12*2*2*2
                                          4*2*4*2       2*2*12*2
                                          4*4*2*2
                                          8*2*2*2
                                          2*2*2*2*2*2

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Positions of 0's are A005117 \ {2}.
The restriction to powers of 2 is A027306.
Heinz numbers of partitions of this type are A028260 /\ A347457.
Positions of 3's appear to be A030514.
Positions of 1's are 1 and A082293.
Allowing non-integer alternating product gives A174725, unordered A339846.
The odd-length version is A347049.
The unordered version is A347438, reverse A347439.
Allowing any length gives A347463.
Partitions of this type are counted by A347704, reverse A035363.
A001055 counts factorizations (strict A045778, ordered A074206).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A339890 counts odd-length factorizations, ordered A174726.
A347050 = factorizations with alternating permutation, complement A347706.
A347437 = factorizations with integer alternating product, reverse A347442.
A347446 = partitions with integer alternating product, reverse A347445.
A347460 counts possible alternating products of factorizations.
Cf. A025047, A038548, A116406, A138364, A347440, A347441, A347454, A347456, A347458, A347459, A347464.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[ordfacs[n],EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]

A347048(n, m=n, ap=1, e=0) = if(1==n,!(e%2) && 1==numerator(ap), sumdiv(n, d, if(d>1, A347048(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

a(n) = A347463(n) - A347049(n).

Data section extended up to a(105) by Antti Karttunen, Jul 28 2024

A347049 Number of odd-length ordered factorizations of n with integer alternating product.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 1, 7, 1, 1, 1, 11, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 14, 1, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 7, 1, 1, 3, 15, 1, 1, 1, 3, 1, 1, 1, 24, 1, 1, 3, 3, 1, 1, 1, 14, 4, 1, 1, 7, 1, 1, 1, 5, 1, 7, 1, 3, 1, 1, 1, 24, 1, 3, 3, 11

Offset: 1

Gus Wiseman, Oct 10 2021

nonn

An ordered factorization of n is a sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(n) ordered factorizations for n = 2, 8, 12, 16, 24, 32, 36, 48:
  2   8       12      16      24      32          36      48
      2*2*2   2*2*3   2*2*4   2*2*6   2*2*8       2*2*9   2*4*6
              3*2*2   2*4*2   3*2*4   2*4*4       2*3*6   3*2*8
                      4*2*2   4*2*3   4*2*4       2*6*3   3*4*4
                              6*2*2   4*4*2       3*2*6   4*2*6
                                      8*2*2       3*3*4   4*4*3
                                      2*2*2*2*2   3*6*2   6*2*4
                                                  4*3*3   6*4*2
                                                  6*2*3   8*2*3
                                                  6*3*2   12*2*2
                                                  9*2*2   2*2*12
                                                          2*2*2*2*3
                                                          2*2*3*2*2
                                                          3*2*2*2*2

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Positions of 2's appear to be A030078.
Positions of 3's appear to be A054753.
Positions of 1's appear to be A167207.
Allowing non-integer alternating product gives A174726, unordered A339890.
The even-length version is A347048.
The unordered version is A347441, with same reverse version.
The case of partitions is A347444, ranked by A347453.
Allowing any length gives A347463.
A001055 counts factorizations (strict A045778, ordered A074206).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A339846 counts even-length factorizations, ordered A174725.
A347050 = factorizations with alternating permutation, complement A347706.
A347437 = factorizations with integer alternating product, reverse A347442.
A347438 = factorizations with alternating product 1, on squares A273013.
A347439 = factorizations with integer reciprocal alternating product.
A347446 = partitions with integer alternating product, reverse A347445.
A347457 lists Heinz numbers of partitions with integer alternating product.
A347460 counts possible alternating products of factorizations.
A347708 counts possible alternating products of odd-length factorizations.
Cf. A025047, A035363, A038548, A116406, A347440, A347454, A347456, A347458, A347459, A347464, A347704.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[ordfacs[n],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]

A347049(n, m=n, ap=1, e=0) = if(1==n,(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1, A347049(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

a(n) = A347463(n) - A347048(n).

Data section extended up to a(100) by Antti Karttunen, Jul 28 2024

A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 14 2021

nonn

This is also the number of distinct possible alternating products of length-3 factorizations of n, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)), and where a factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			Representative factorizations for each of the a(360) = 9 alternating products:
   (2,2,90) -> 90
   (2,3,60) -> 40
   (2,4,45) -> 45/2
   (2,5,36) -> 72/5
   (2,6,30) -> 10
   (2,9,20) -> 40/9
  (2,10,18) -> 18/5
  (2,12,15) -> 5/2
   (3,8,15) -> 45/8

Antti Karttunen, Table of n, a(n) for n = 1..65537

Allowing factorizations of any length <= 3 gives A033273.
Positions of positive terms are A033942.
Positions of 0's are A037143.
The length-2 version is A072670.
Allowing any length gives A347460, reverse A038548.
Allowing any odd length gives A347708.
A001055 counts factorizations (strict A045778, ordered A074206).
A122179 counts length-3 factorizations.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions, positive A276024.
Cf. A000040, A001358, A002033, A046951, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456, A347461.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],Length[#]==3&]]],{n,100}]

A347709(n) = { my(rats=List([])); fordiv(n,z,my(yx=n/z); fordiv(yx, y, my(x = yx/y); if((y <= z) && (x <= y) && (x > 1), listput(rats,x*z/y)))); #Set(rats); }; \\ Antti Karttunen, Jan 29 2025

More terms from Antti Karttunen, Jan 29 2025

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A348381 Number of inseparable factorizations of n that are not a twin (x*x).

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A348383 Number of factorizations of n that are either separable (have an anti-run permutation) or are a twin (x*x).

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A347447 Number of strict factorizations of n with alternating product > 1.

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A347708 Number of distinct possible alternating products of odd-length factorizations of n.

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A347048 Number of even-length ordered factorizations of n with integer alternating product.

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A347049 Number of odd-length ordered factorizations of n with integer alternating product.

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A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

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