A347438 -id:A347438 - cp's OEIS frontend

A348609 Numbers with a separable factorization without an alternating permutation.

216, 270, 324, 378, 432, 486, 540, 594, 640, 648, 702, 756, 768, 810, 864, 896, 918, 960, 972, 1024, 1026, 1080, 1134, 1152, 1188, 1242, 1280, 1296, 1344, 1350, 1404, 1408, 1458, 1500, 1512, 1536, 1566, 1620, 1664, 1674, 1728, 1750, 1782, 1792, 1836, 1890

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

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.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of sets.
Note that 216 has separable prime factorization (2*2*2*3*3*3) with an alternating permutation, but the separable factorization (2*3*3*3*4) is has no alternating permutation. See also A345173.

			The terms and their prime factorizations begin:
  216 = 2*2*2*3*3*3
  270 = 2*3*3*3*5
  324 = 2*2*3*3*3*3
  378 = 2*3*3*3*7
  432 = 2*2*2*2*3*3*3
  486 = 2*3*3*3*3*3
  540 = 2*2*3*3*3*5
  594 = 2*3*3*3*11
  640 = 2*2*2*2*2*2*2*5
  648 = 2*2*2*3*3*3*3
  702 = 2*3*3*3*13
  756 = 2*2*3*3*3*7
  768 = 2*2*2*2*2*2*2*2*3
  810 = 2*3*3*3*3*5
  864 = 2*2*2*2*2*3*3*3

Partitions of this type are counted by A345166, ranked by A345173 (a superset).
Compositions of this type are counted by A345195, ranked by A345169.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A025047 counts alternating compositions, complement A345192, ranked by A345167.
A335434 counts separable factorizations, with twins A348383, complement A333487.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions w/o an alternating permutation, complement A345170.
A347438 counts factorizations with alternating product 1, additive A119620.
A348379 counts factorizations w/ an alternating permutation, complement A348380.
A348610 counts alternating ordered factorizations, complement A348613.
Cf. A001248, A003242, A038548, A325534, A336103, A344614, A345162, A347050, A347437, A347706, A348377, A348381, 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_,_}]&]!={};
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[1000],Function[n,Select[facs[n],sepQ[#]&&Select[Permutations[#],wigQ]=={}&]!={}]]

A347459 Number of factorizations of n^2 with integer reciprocal alternating product.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 6, 3, 4, 1, 11, 1, 4, 4, 12, 1, 11, 1, 12, 4, 4, 1, 28, 3, 4, 6, 12, 1, 19, 1, 22, 4, 4, 4, 38, 1, 4, 4, 29, 1, 21, 1, 12, 11, 4, 1, 65, 3, 11, 4, 12, 1, 29, 4, 29, 4, 4, 1, 71, 1, 4, 11, 40, 4, 22, 1, 12, 4, 18, 1, 107, 1, 4, 11, 12, 4, 22, 1, 66, 12, 4, 1, 76, 4, 4, 4, 30, 1, 71, 4, 12, 4, 4, 4, 141

Offset: 1

Gus Wiseman, Sep 22 2021

nonn

We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
All such factorizations have even length.
Image appears to be 1, 3, 4, 6, 11, ... , missing some numbers such as 2, 5, 7, 8, 9, ...
The case of alternating product 1, the case of alternating sum 0, and the reverse version are all counted by A001055.

			The a(2) = 1 through a(10) = 4 factorizations:
    2*2  3*3  2*8      5*5  6*6      7*7  8*8          9*9      2*50
              4*4           2*18          2*32         3*27     5*20
              2*2*2*2       3*12          4*16         3*3*3*3  10*10
                            2*2*3*3       2*2*2*8               2*2*5*5
                                          2*2*4*4
                                          2*2*2*2*2*2

Antti Karttunen, Table of n, a(n) for n = 1..16384

Positions of 1's are 1 and A000040, squares A001248.
The additive version (partitions) is A000041, the even bisection of A119620.
Partitions of this type are ranked by A028982 and A347451.
The restriction to powers of 2 is A236913, the even bisection of A027187.
The nonsquared nonreciprocal even-length version is A347438.
This is the restriction to perfect squares of A347439.
The nonreciprocal version is A347458, non-squared A347437.
A000290 lists squares, complement A000037.
A001055 counts factorizations.
A046099 counts factorizations with no alternating permutations.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347457 ranks partitions with integer alternating product.
A347466 counts factorizations of n^2.
Cf. A062312, A316523, A330972, A344653, A346635, A347440, A347441, A347442, A347453, A347461, A347463, A347464, A347705.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
recaltprod[q_]:=Product[q[[i]]^(-1)^i,{i,Length[q]}];
Table[Length[Select[facs[n^2],IntegerQ[recaltprod[#]]&]],{n,100}]

A347439(n, m=n, ap=1, e=0) = if(1==n, !(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1 && d<=m, A347439(n/d, d, ap * d^((-1)^e), 1-e))));
A347459(n) = A347439(n^2); \\ Antti Karttunen, Jul 28 2024

a(2^n) = A236913(n).

a(n) = A347439(n^2).

Data section extended up to a(96) by Antti Karttunen, Jul 28 2024

A347705 Number of factorizations of n with reverse-alternating product > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 1, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 8, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 3, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 12 2021

nonn

A factorization of n is a weakly increasing 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 reverse-alternating product is the alternating product of the reversed sequence.

			The a(n) factorizations for n = 2, 6, 8, 12, 24, 30, 48, 60:
  2   6     8       12      24        30      48          60
      2*3   2*4     2*6     3*8       5*6     6*8         2*30
            2*2*2   3*4     4*6       2*15    2*24        3*20
                    2*2*3   2*12      3*10    3*16        4*15
                            2*2*6     2*3*5   4*12        5*12
                            2*3*4             2*3*8       6*10
                            2*2*2*3           2*4*6       2*5*6
                                              3*4*4       3*4*5
                                              2*2*12      2*2*15
                                              2*2*2*6     2*3*10
                                              2*2*3*4     2*2*3*5
                                              2*2*2*2*3

Positions of 1's are A000430.
The weak version (>= instead of >) is A001055, non-reverse A347456.
The non-reverse version is A339890, strict A347447.
The version for reverse-alternating product 1 is A347438.
Allowing any integer reciprocal alternating product gives A347439.
The even-length case is A347440, also the opposite reverse version.
Allowing any integer rev-alt product gives A347442, non-reverse A347437.
The version for partitions is A347449, non-reverse A347448.
A001055 counts factorizations (strict A045778, ordered A074206).
A038548 counts possible rev-alt products of factorizations, integer A046951.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A292886 counts knapsack factorizations, by sum A293627.
A347707 counts possible integer reverse-alternating products of partitions.
Cf. A028983, A119620, A339846, A347441, A347443, A347450, A347463, A347466.

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

a(n) = A001055(n) - A347438(n).

A347464 Number of even-length ordered factorizations of n^2 into factors > 1 with alternating product 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 1, 6, 2, 5, 1, 26, 1, 5, 5, 20, 1, 26, 1, 26, 5, 5, 1, 134, 2, 5, 6, 26, 1, 73, 1, 70, 5, 5, 5, 230, 1, 5, 5, 134, 1, 73, 1, 26, 26, 5, 1, 670, 2, 26, 5, 26, 1, 134, 5, 134, 5, 5, 1, 686, 1, 5, 26, 252, 5, 73, 1, 26, 5, 73, 1, 1714, 1, 5, 26

Offset: 1

Gus Wiseman, Sep 23 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)).
Also the number of ordered pairs of ordered factorizations of n, both of the same length.
Note that the version for all n (not just squares) is 0 except at perfect squares.

			The a(12) = 26 ordered factorizations:
  (2*2*6*6)      (3*2*4*6)      (6*2*2*6)  (4*2*3*6)  (12*12)
  (2*3*6*4)      (3*3*4*4)      (6*3*2*4)  (4*3*3*4)
  (2*4*6*3)      (3*4*4*3)      (6*4*2*3)  (4*4*3*3)
  (2*6*6*2)      (3*6*4*2)      (6*6*2*2)  (4*6*3*2)
  (2*2*2*2*3*3)  (3*2*2*2*2*3)
  (2*2*2*3*3*2)  (3*2*2*3*2*2)
  (2*2*3*2*2*3)  (3*3*2*2*2*2)
  (2*2*3*3*2*2)
  (2*3*2*2*3*2)
  (2*3*3*2*2*2)
For example, the ordered factorization 6*3*2*4 = 144 has alternating product 6/3*2/4 = 1, so is counted under a(12).

Index entries for sequences computed from exponents in factorization of n

Positions of 1's are A008578 (1 and A000040).
The restriction to powers of 2 is A000984.
Positions of 2's are A001248.
The not necessarily even-length version is A273013.
A000290 lists squares, complement A000037.
A001055 counts factorizations.
A027187 counts even-length partitions.
A074206 counts ordered factorizations.
A119620 counts partitions with alternating product 1, ranked by A028982.
A339846 counts even-length factorizations, ordered A347706.
A347438 counts factorizations with alternating product 1.
A347457 ranks partitions with integer alternating product.
A347460 counts possible alternating products of factorizations.
A347466 counts factorizations of n^2.
Cf. A062312, A339890, A347437, A347439, A347440, A347442, A347456, A347459, A347463, A347705.

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[Join@@Permutations/@facs[n^2],EvenQ[Length[#]]&&altprod[#]==1&]],{n,100}]

A347464aux(n, k=0, t=1) = if(1==n, (0==k)&&(1==t), my(s=0); fordiv(n, d, if((d>1), s += A347464aux(n/d, 1-k, t*(d^((-1)^k))))); (s));
A347464(n) = A347464aux(n^2); \\ Antti Karttunen, Oct 30 2021

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

Original entry on oeis.org

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).

A347466 Number of factorizations of n^2.

Original entry on oeis.org

1, 2, 2, 5, 2, 9, 2, 11, 5, 9, 2, 29, 2, 9, 9, 22, 2, 29, 2, 29, 9, 9, 2, 77, 5, 9, 11, 29, 2, 66, 2, 42, 9, 9, 9, 109, 2, 9, 9, 77, 2, 66, 2, 29, 29, 9, 2, 181, 5, 29, 9, 29, 2, 77, 9, 77, 9, 9, 2, 269, 2, 9, 29, 77, 9, 66, 2, 29, 9, 66, 2, 323, 2, 9, 29, 29

Offset: 1

Gus Wiseman, Sep 23 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			The a(1) = 1 through a(8) = 11 factorizations:
  ()  (4)    (9)    (16)       (25)   (36)       (49)   (64)
      (2*2)  (3*3)  (2*8)      (5*5)  (4*9)      (7*7)  (8*8)
                    (4*4)             (6*6)             (2*32)
                    (2*2*4)           (2*18)            (4*16)
                    (2*2*2*2)         (3*12)            (2*4*8)
                                      (2*2*9)           (4*4*4)
                                      (2*3*6)           (2*2*16)
                                      (3*3*4)           (2*2*2*8)
                                      (2*2*3*3)         (2*2*4*4)
                                                        (2*2*2*2*4)
                                                        (2*2*2*2*2*2)

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

Positions of 2's are the primes (A000040), which have squares A001248.
The restriction to powers of 2 is A058696.
The additive version (partitions) is A072213.
The case of integer alternating product is A347459, nonsquared A347439.
A000290 lists squares, complement A000037.
A001055 counts factorizations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347050 = factorizations with alternating permutation, complement A347706.
Cf. A000041, A062312, A120452, A144338, A273013, A330972, A345957, A346635, A347437, A347438, A347457, A347460, A347464.

b:= proc(n, k) option remember; `if`(n>k, 0, 1)+`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=numtheory[divisors](n) minus {1, n}))
    end:
a:= proc(n) option remember; b((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
      sort(map(i-> i[2], ifactors(n^2)[2]), `>`))$2)
    end:
seq(a(n), n=1..76);  # Alois P. Heinz, Oct 14 2021

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[n^2]],{n,25}]

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));
A347466(n) = A001055(n^2); \\ Antti Karttunen, Oct 13 2021

a(n) = A001055(A000290(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.

A348611 Number of ordered factorizations of n with no adjacent equal factors.

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, 14, 1, 3, 3, 6, 1, 13, 1, 7, 3, 3, 3, 17, 1, 3, 3, 14, 1, 13, 1, 6, 6, 3, 1, 29, 1, 6, 3, 6, 1, 14, 3, 14, 3, 3, 1, 36, 1, 3, 6, 14, 3, 13, 1, 6, 3, 13, 1, 45, 1, 3, 6, 6, 3, 13, 1, 29, 4, 3

Offset: 1

Gus Wiseman, Nov 07 2021

nonn

First differs from A348610 at a(24) = 14, A348610(24) = 12.
An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
In analogy with Carlitz compositions, these may be called Carlitz ordered factorizations.

			The a(n) ordered factorizations without adjacent equal factors for n = 1, 6, 12, 16, 24, 30, 32, 36 are:
  ()   6     12      16      24      30      32      36
       2*3   2*6     2*8     3*8     5*6     4*8     4*9
       3*2   3*4     8*2     4*6     6*5     8*4     9*4
             4*3     2*4*2   6*4     10*3    16*2    12*3
             6*2             8*3     15*2    2*16    18*2
             2*3*2           12*2    2*15    2*8*2   2*18
                             2*12    3*10    4*2*4   3*12
                             2*3*4   2*3*5           2*3*6
                             2*4*3   2*5*3           2*6*3
                             2*6*2   3*2*5           2*9*2
                             3*2*4   3*5*2           3*2*6
                             3*4*2   5*2*3           3*4*3
                             4*2*3   5*3*2           3*6*2
                             4*3*2                   6*2*3
                                                     6*3*2
                                                     2*3*2*3
                                                     3*2*3*2
Thus, of total A074206(12) = 8 ordered factorizations of 12, only factorizations 2*2*3 and 3*2*2 (see A348616) are not included in this count, therefore a(12) = 6. - _Antti Karttunen_, Nov 12 2021

The additive version (compositions) is A003242, complement A261983.
The additive alternating version is A025047, ranked by A345167.
Factorizations without a permutation of this type are counted by A333487.
As compositions these are ranked by A333489, complement A348612.
Factorizations with a permutation of this type are counted by A335434.
The non-alternating additive version is A345195, ranked by A345169.
The alternating case is A348610, which is dominated at positions A122181.
The complement is counted by A348616.
A001055 counts factorizations, strict A045778, ordered A074206.
A325534 counts separable partitions, ranked by A335433.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A348613 counts non-alternating ordered factorizations.
Cf. A001250, A138364, A336103, A347050, A347438, A347463, A347466, A347706, A348379, A348383.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n],antirunQ]],{n,100}]

A348611(n, e=0) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d!=e), s += A348611(n/d, d))); (s)); \\ Antti Karttunen, Nov 12 2021

a(n) = A074206(n) - A348616(n).

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

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