A347460 -id:A347460 - cp's OEIS frontend

A347437 Number of factorizations of n with integer alternating product.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 4, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 6, 1, 2, 2, 6, 1, 1, 1, 2, 1, 1, 1, 7

Offset: 1

Gus Wiseman, Sep 06 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 factorizations for n = 4, 16, 36, 48, 54, 64, 108:
  (4)   (16)      (36)      (48)        (54)    (64)          (108)
  (2*2) (4*4)     (6*6)     (2*4*6)     (2*3*9) (8*8)         (2*6*9)
        (2*2*4)   (2*2*9)   (3*4*4)     (3*3*6) (2*4*8)       (3*6*6)
        (2*2*2*2) (2*3*6)   (2*2*12)            (4*4*4)       (2*2*27)
                  (3*3*4)   (2*2*2*2*3)         (2*2*16)      (2*3*18)
                  (2*2*3*3)                     (2*2*4*4)     (3*3*12)
                                                (2*2*2*2*4)   (2*2*3*3*3)
                                                (2*2*2*2*2*2)

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

Positions of 1's are A005117, complement A013929.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The restriction to powers of 2 is A344607.
The even-length case is A347438, also the case of alternating product 1.
The reciprocal version is A347439.
Allowing any alternating product < 1 gives A347440.
The odd-length case is A347441.
The reverse version is A347442.
The additive version is A347446, ranked by A347457.
Allowing any alternating product >= 1 gives A347456.
The restriction to perfect squares is A347458, reciprocal A347459.
The ordered version is A347463.
A001055 counts factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors of n (reverse: A071322).
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
Cf. A025047, A038548, A062312, A088218, A119620, A316523, A330972, A332269, A347445, A347447, A347451, A347454.

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],IntegerQ@*altprod]],{n,100}]

A347437(n, m=n, ap=1, e=0) = if(1==n, if(e%2, 1==denominator(ap), 1==numerator(ap)), sumdiv(n, d, if((d>1)&&(d<=m), A347437(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(2^n) = A344607(n).

a(n^2) = A347458(n).

Data section extended up to a(108) by Antti Karttunen, Oct 22 2023

A347438 Number of unordered factorizations of n with alternating product 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 06 2021

nonn

Also the number of unordered factorizations of n with alternating sum 0.
Also the number of unordered factorizations of n with all even multiplicities.
This is the even-length case of A347437, the odd-length case being A347441.
An unordered 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 a(n) factorizations for n = 16, 64, 144, 256, 576:
  4*4      8*8          12*12        16*16            24*24
  2*2*2*2  2*2*4*4      2*2*6*6      2*2*8*8          3*3*8*8
           2*2*2*2*2*2  3*3*4*4      4*4*4*4          4*4*6*6
                        2*2*2*2*3*3  2*2*2*2*4*4      2*2*12*12
                                     2*2*2*2*2*2*2*2  2*2*2*2*6*6
                                                      2*2*3*3*4*4
                                                      2*2*2*2*2*2*3*3

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

Positions of zeros are A000037.
Positions of nonzero terms are A000290.
The restriction to perfect squares is A001055 (ordered: A273013).
The restriction to powers of 2 is A035363.
The additive version is A119620, ranked by A028982.
Positions of non-1's are A213367 \ {1}.
Positions of 1's are A280076 = {1} \/ A001248.
Sorted first positions are 1, 2, and all terms of A330972 squared.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
Allowing any integer alternating product gives A347437.
Allowing any integer reciprocal alternating product gives A347439.
Allowing any alternating product < 1 gives A347440.
Allowing any alternating product >= 1 gives A347456.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime factors.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A000041, A005117, A025047, A038548, A062312, A088218, A316523, A332269, A344607, A347442, A347446, 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],altprod[#]==1&]],{n,100}]

A347438(n, m=n, k=0, t=1) = if(1==n, (1==t), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A347438(n/d, d, 1-k, t*(d^((-1)^k))))); (s)); \\ Antti Karttunen, Oct 30 2021

a(2^n) = A035363(n).

a(n^2) = A001055(n).

Name and comments clarified (with unordered) by Jacob Sprittulla, Oct 05 2021

A347439 Number of factorizations of n with integer reciprocal 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, 3, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 6, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 1, 4, 0, 0, 0, 1, 0, 0, 0, 5

Offset: 1

Gus Wiseman, Sep 07 2021

nonn

All of these factorizations have an even number of factors, so their reverse-alternating product is also an integer.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
The value of a(n) does not depend solely on the prime signature of n. See the example comparing a(144) and a(400). - Antti Karttunen, Jul 28 2024

			The a(n) factorizations for
n    = 16,       36,       64,           72,       128,          144:
a(n) = 3,        4,        6,            5,        7,            11
--------------------------------------------------------------------------------
       2*8       6*6       8*8           2*36      2*64          2*72
       4*4       2*18      2*32          3*24      4*32          3*48
       2*2*2*2   3*12      4*16          6*12      8*16          4*36
                 2*2*3*3   2*2*2*8       2*2*3*6   2*2*4*8       6*24
                           2*2*4*4       2*3*3*4   2*4*4*4       12*12
                           2*2*2*2*2*2             2*2*2*16      2*2*6*6
                                                   2*2*2*2*2*4   2*3*3*8
                                                                 3*3*4*4
                                                                 2*2*2*18
                                                                 2*2*3*12
                                                                 2*2*2*2*3*3
From _Antti Karttunen_, Jul 28 2024 (Start)
For n=400, there are 12 such factorizations:
  2*200
  4*100
  5*80
  10*40
  20*20
  2*2*2*50
  2*2*5*20
  2*2*10*10
  2*4*5*10
  2*5*5*8
  4*4*5*5
  2*2*2*2*5*5.
Note that 400 = 2^4 * 5^2 has the same prime signature as 144 = 2^4 * 3^2. 400 = 2*4*5*10 is the factorization for which there is no analogous factorization of 144, as 2*3*4*6 doesn't satisfy the condition of having an integer reciprocal alternating product.
(End)

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

Positions of 0's are A005117 \ {1}.
Positions of non-0's are 1 and A013929.
The restriction to powers of 2 is A027187, reverse A035363.
Positions of 1's are 1 and A082293.
The additive version is A119620, ranked by A347451 and A028982.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The non-reciprocal version is A347437.
The reverse version is A347438.
Allowing any alternating product < 1 gives A347440.
The non-reciprocal reverse version is A347442.
Allowing any alternating product >= 1 gives A347456.
The restriction to perfect squares is A347459, non-reciprocal A347458.
A038548 counts possible reverse-alternating products of factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A236913, A316523, A330972, A332269, A344606, A344607, A347445, A347446, A347454, A347457, A347463.

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],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)))); \\ Antti Karttunen, Jul 28 2024

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

a(2^n) = A027187(n).

a(n^2) = A347459(n).

Data section extended up to a(108) by Antti Karttunen, Jul 28 2024

A347442 Number of factorizations of n with integer reverse-alternating product.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 3, 2, 1, 3, 3, 1, 1, 1, 7, 1, 1, 1, 8, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 8, 2, 3, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 11, 1, 1, 3, 3, 1, 1, 1, 8, 5, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 9, 1, 3, 3, 8, 1, 1, 1, 3, 1, 1, 1, 12

Offset: 1

Gus Wiseman, Sep 08 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 = 4, 8, 16, 32, 36, 54, 64:
  (4)    (8)      (16)       (32)         (36)       (54)     (64)
  (2*2)  (2*4)    (2*8)      (4*8)        (6*6)      (3*18)   (8*8)
         (2*2*2)  (4*4)      (2*16)       (2*18)     (2*3*9)  (2*32)
                  (2*2*4)    (2*2*8)      (3*12)     (3*3*6)  (4*16)
                  (2*2*2*2)  (2*4*4)      (2*2*9)             (2*4*8)
                             (2*2*2*4)    (2*3*6)             (4*4*4)
                             (2*2*2*2*2)  (3*3*4)             (2*2*16)
                                          (2*2*3*3)           (2*2*2*8)
                                                              (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..65537

The restriction to powers of 2 is A000041, reverse A344607.
Positions of 2's are A001248.
Positions of 1's are A005117.
Positions of non-1's are A013929.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The non-reverse version is A347437.
The reciprocal version is A347438.
The even-length case is A347439.
Allowing any alternating product < 1 gives A347440.
The odd-length case is A347441, ranked by A347453.
The additive version is A347445, ranked by A347457.
The non-reverse additive version is A347446, ranked by A347454.
Allowing any alternating product >= 1 gives A347456.
The ordered version is A347463.
A038548 counts possible reverse-alternating products of factorizations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A273013 counts ordered factorizations of n^2 with alternating product 1.
Cf. A000290, A025047, A330972, A347443, A347449, A347451, A347458, A347459, A347460, A347462.

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

A347442(n, m=n, ap=1, e=0) = if(1==n, 1==denominator(ap), sumdiv(n, d, if((d>1)&&(d<=m), A347442(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(2^n) = A000041(n).

Data section extended up to a(108) by Antti Karttunen, Oct 22 2023

A347463 Number of ordered factorizations of n with integer alternating product.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 7, 1, 4, 1, 4, 1, 1, 1, 6, 2, 1, 3, 4, 1, 1, 1, 11, 1, 1, 1, 18, 1, 1, 1, 6, 1, 1, 1, 4, 4, 1, 1, 20, 2, 4, 1, 4, 1, 6, 1, 6, 1, 1, 1, 8, 1, 1, 4, 26, 1, 1, 1, 4, 1, 1, 1, 35, 1, 1, 4, 4, 1, 1, 1, 20, 7, 1, 1, 8, 1, 1, 1, 6, 1, 8, 1, 4, 1, 1, 1, 32, 1, 4, 4, 18

Offset: 1

Gus Wiseman, Oct 07 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 ordered factorizations for n = 4, 8, 12, 16, 24, 32, 36:
  4     8       12      16        24      32          36
  2*2   4*2     6*2     4*4       12*2    8*4         6*6
        2*2*2   2*2*3   8*2       2*2*6   16*2        12*3
                3*2*2   2*2*4     3*2*4   2*2*8       18*2
                        2*4*2     4*2*3   2*4*4       2*2*9
                        4*2*2     6*2*2   4*2*4       2*3*6
                        2*2*2*2           4*4*2       2*6*3
                                          8*2*2       3*2*6
                                          2*2*4*2     3*3*4
                                          4*2*2*2     3*6*2
                                          2*2*2*2*2   4*3*3
                                                      6*2*3
                                                      6*3*2
                                                      9*2*2
                                                      2*2*3*3
                                                      2*3*3*2
                                                      3*2*2*3
                                                      3*3*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 A001248.
Positions of 1's are A005117.
The restriction to powers of 2 is A116406.
The even-length case is A347048
The odd-length case is A347049.
The unordered version is A347437, reciprocal A347439, reverse A347442.
The case of partitions is A347446, reverse A347445, ranked by A347457.
A001055 counts factorizations (strict A045778, ordered A074206).
A046099 counts factorizations with no alternating permutations.
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.
A339846 counts even-length factorizations, ordered A174725.
A339890 counts odd-length factorizations, ordered A174726.
A347438 counts factorizations with alternating product 1.
A347460 counts possible alternating products of factorizations.
Cf. A025047, A038548, A138364, A347440, A347441, A347453, A347454, A347456, A347458, A347459, A347464, A347705, A347708.

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],IntegerQ[altprod[#]]&]],{n,100}]

A347463(n, m=n, ap=1, e=0) = if(1==n, if(e%2, 1==denominator(ap), 1==numerator(ap)), sumdiv(n, d, if(d>1, A347463(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

a(n) = A347048(n) + A347049(n).

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

A347440 Number of factorizations of n with alternating product < 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 4, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 1, 3, 0, 1, 1, 4, 0, 3, 0, 2, 2, 1, 0, 6, 0, 2, 1, 2, 0, 4, 1, 4, 1, 1, 0, 6, 0, 1, 2, 3, 1, 3, 0, 2, 1, 3, 0, 8, 0, 1, 2, 2, 1, 3, 0, 6, 1, 1, 0, 6, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 07 2021

nonn

All such factorizations have even length and alternating sum < 0, so partitions of this type are counted by A344608.
Also the number of factorizations of n with alternating sum < 0.
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 a(n) factorizations for n = 6, 12, 24, 30, 48, 72, 96, 120:
  2*3  2*6  3*8      5*6   6*8      8*9      2*48         2*60
       3*4  4*6      2*15  2*24     2*36     3*32         3*40
            2*12     3*10  3*16     3*24     4*24         4*30
            2*2*2*3        4*12     4*18     6*16         5*24
                           2*2*2*6  6*12     8*12         6*20
                           2*2*3*4  2*2*2*9  2*2*3*8      8*15
                                    2*2*3*6  2*2*4*6      10*12
                                    2*3*3*4  2*3*4*4      2*2*5*6
                                             2*2*2*12     2*3*4*5
                                             2*2*2*2*2*3  2*2*2*15
                                                          2*2*3*10

PlanetMath, alternating sum

Positions of 0's are A000430.
Positions of 2's are A054753.
Positions of non-0's are A080257.
Positions of 1's are A332269.
The weak version (<= 1 instead of < 1) is A339846, ranked by A028982.
The reciprocal version is A339890.
The additive version is A344608, ranked by A119899.
The even-sum additive version is A344743, ranked by A119899 /\ A300061.
Allowing any integer alternating product gives A347437, additive A347446.
The equal version (= 1 instead of < 1) is A347438.
Allowing any integer reciprocal alternating product gives A347439.
The complement (>= 1 instead of < 1) is counted by A347456.
A038548 counts possible reverse-alternating products of factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
Cf. A008549, A058622, A119620, A294175, A330972, A347442, A347447, A347454.

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],altprod[#]<1&]],{n,100}]

a(2^n) = A344608(n).

a(n) = A339846(n) - A347438(n).

A273013 Number of different arrangements of nonnegative integers on a pair of n-sided dice such that the dice can add to any integer from 0 to n^2-1.

Original entry on oeis.org

1, 1, 1, 3, 1, 7, 1, 10, 3, 7, 1, 42, 1, 7, 7, 35, 1, 42, 1, 42, 7, 7, 1, 230, 3, 7, 10, 42, 1, 115, 1, 126, 7, 7, 7, 393, 1, 7, 7, 230, 1, 115, 1, 42, 42, 7, 1, 1190, 3, 42, 7, 42, 1, 230, 7, 230, 7, 7, 1, 1158, 1, 7, 42, 462, 7, 115, 1, 42, 7, 115, 1, 3030

Offset: 1

Elliott Line, May 13 2016

The set of b values (see formula), and therefore also a(n), depends only on the prime signature of n. So, for example, a(24) will be identical to a(n) of any other n which is also of the form p_1^3*p_2, (e.g., 40, 54, 56).
The value of b_1 will always be 1. When n is prime, the only nonzero b will be b_1, so therefore a(n) will be 1.
In any arrangement, both dice will have a 0, and one will have a 1 (here called the lead die). To determine any one of the actual arrangements to numbers on the dice, select one of the permutations of divisors (for the lead die), then select another permutation that is either the same length as that of the lead die, or one less. For example, if n = 12, we might select 2*3*2 for the lead die, and 3*4 for the second die. These numbers effectively tell you when to "switch track" when numbering the dice, and will uniquely result in the numbering: (0,1,6,7,12,13,72,73,78,79,84,85; 0,2,4,18,20,22,36,38,40,54,56,58).
a(n) is the number of (unordered) pairs of polynomials c(x) = x^c_1 + x^c_2 + ... + x^c_n, d(x) = x^d_1 + x^d_2 + ... + x^d_n with nonnegative integer exponents such that c(x)*d(x) = (x^(n^2)-1)/(x-1). - Alois P. Heinz, May 13 2016
a(n) is also the number of principal reversible squares of order n. - S. Harry White, May 19 2018
From Gus Wiseman, Oct 29 2021: (Start)
Also the number of ordered factorizations of n^2 with alternating product 1. This follows from the author's formula. Taking n instead of n^2 gives a(sqrt(n)) if n is a perfect square, otherwise 0. Here, an ordered factorization of n is a sequence of positive integers > 1 with product n, and the alternating product of a sequence (y_1,...,y_k) is Product_i y_i^((-1)^(i-1)). For example, the a(1) = 1 through a(9) = 3 factorizations are:
  ()  (22)  (33)  (44)    (55)  (66)    (77)  (88)      (99)
                  (242)         (263)         (284)     (393)
                  (2222)        (362)         (482)     (3333)
                                (2233)        (2244)
                                (2332)        (2442)
                                (3223)        (4224)
                                (3322)        (4422)
                                              (22242)
                                              (24222)
                                              (222222)
The even-length case is A347464.
(End)

			When n = 4, a(n) = 3; the three arrangements are (0,1,2,3; 0,4,8,12), (0,1,4,5; 0,2,8,10), (0,1,8,9; 0,2,4,6).
When n = 5, a(n) = 1; the sole arrangement is (0,1,2,3,4; 0,5,10,15,20).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Matthew C. Lettington and Karl Michael Schmidt, Divisor Functions and the Number of Sum Systems, arXiv:1910.02455 [math.NT], 2019.
S. Harry White, Reversible Squares

Cf. A111588, A131514, A360098.
Positions of 1's are 1 and A000040.
A000290 lists squares, complement A000037.
A001055 counts factorizations, ordered A074206.
A119620 counts partitions with alternating product 1, ranked by A028982.
A339846 counts even-length factorizations, ordered A174725.
A339890 counts odd-length factorizations, ordered A174726.
A347438 counts factorizations with alternating product 1.
A347460 counts possible alternating products of factorizations.
A347463 counts ordered factorizations with integer alternating product.
A347466 counts factorizations of n^2.
Cf. A062312, A347437, A347439, A347440, A347442, A347456, A347459, A347464.

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], altprod[#] == 1&]],{n, 30}]
(* Gus Wiseman, Oct 29 2021 *)
(* or *)
ofc[n_,k_] := If[k > PrimeOmega[n], 0, If[k == 0 && n == 1, 1, Sum[ofc[n/d, k-1],{d, Rest[Divisors[n]]}]]];
Table[If[n == 1, 1, Sum[ofc[n, x]^2 + ofc[n, x]*ofc[n, x+1], {x, n}]],{n, 30}]
(* Gus Wiseman, Oct 29 2021, based on author's formula *)

A273013aux(n, k=0, t=1) = if(1==n, (1==t), my(s=0); fordiv(n, d, if((d>1), s += A273013aux(n/d, 1-k, t*(d^((-1)^k))))); (s));
A273013(n) = A273013aux(n^2); \\ Antti Karttunen, Oct 30 2021

@cached_function
def r(m,n):
    if n==1:
        return(1)
    divList = divisors(m)[:-1]
    return(sum(r(n,d) for d in divList))
def A273013(n):
    return(r(n,n)) # William P. Orrick, Feb 19 2023

a(n) = b_1^2 + b_2^2 + b_3^2 + ... + b_1*b_2 + b_2*b_3 + b_3*b_4 + ..., where b_k is the number of different permutations of k divisors of n to achieve a product of n. For example, for n=24, b_3 = 9 (6 permutations of 2*3*4 and 3 permutations of 2*2*6).

a(n) = r(n,n) where r(m,1) = 1 and r(m,n) = Sum_{d|m,dWilliam P. Orrick, Feb 19 2023

A347461 Number of distinct possible alternating products of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 10, 12, 16, 19, 23, 27, 34, 41, 49, 57, 67, 78, 91, 106, 125, 147, 166, 187, 215, 245, 277, 317, 357, 405, 460, 524, 592, 666, 740, 829, 928, 1032, 1147, 1273, 1399, 1555, 1713, 1892, 2087, 2298, 2523, 2783, 3070, 3383, 3724, 4104, 4504

Offset: 0

Gus Wiseman, Oct 06 2021

nonn

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

			Partitions representing each of the a(7) = 10 alternating products are:
     (7) -> 7
    (61) -> 6
    (52) -> 5/2
   (511) -> 5
    (43) -> 4/3
   (421) -> 2
  (4111) -> 4
   (331) -> 1
   (322) -> 3
  (3211) -> 3/2

The version for alternating sum is A004526.
Counting only integers gives A028310, reverse A347707.
The version for factorizations is A347460, reverse A038548.
The reverse version is A347462.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A122768 counts distinct submultisets of partitions.
A126796 counts complete partitions.
A293627 counts knapsack factorizations by sum.
A301957 counts distinct subset-products of prime indices.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
A304793 counts distinct positive subset-sums of prime indices.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A001055, A002033, A002219, A028983, A119620, A325768, A345926, A347443, A347444, A347445, A347446.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@IntegerPartitions[n]]],{n,0,30}]

A347441 Number of odd-length factorizations of n with integer alternating product.

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, 2, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 5, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 6, 1, 2, 2, 4, 1, 1, 1, 2, 1, 1, 1, 7

Offset: 1

Gus Wiseman, Sep 07 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 a(n) factorizations for n = 2, 8, 32, 48, 54, 72, 108:
  2   8       32          48          54      72          108
      2*2*2   2*2*8       2*4*6       2*3*9   2*6*6       2*6*9
              2*4*4       3*4*4       3*3*6   3*3*8       3*6*6
              2*2*2*2*2   2*2*12              2*2*18      2*2*27
                          2*2*2*2*3           2*3*12      2*3*18
                                              2*2*2*3*3   3*3*12
                                                          2*2*3*3*3

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

The restriction to powers of 2 is A027193.
Positions of 1's are A167207 = A005117 \/ A001248.
Allowing any alternating product gives A339890.
Allowing even-length factorizations gives A347437.
The even-length instead of odd-length version is A347438.
The additive version is A347444, ranked by A347453.
A038548 counts possible reverse-alternating products of factorizations.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A339846 counts even-length factorizations.
A347439 counts factorizations with integer reciprocal alternating product.
A347440 counts factorizations with alternating product < 1.
A347442 counts factorizations with integer reverse-alternating product.
A347456 counts factorizations with alternating product >= 1.
A347463 counts ordered factorizations with integer alternating product.
Cf. A062312, A119620, A236913, A330972, A347445, A347446, A347447, A347451, A347458, A347460.

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],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]

A347441(n, m=n, ap=1, e=0) = if(1==n, (e%2)&&1==denominator(ap), sumdiv(n, d, if((d>1)&&(d<=m), A347441(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(2^n) = A027193(n).

Data section extended up to a(108) by Antti Karttunen, Oct 22 2023

A347462 Number of distinct possible reverse-alternating products of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 13, 17, 22, 28, 33, 42, 51, 59, 69, 84, 100, 117, 137, 163, 191, 222, 256, 290, 332, 378, 429, 489, 564, 643, 729, 819, 929, 1040, 1167, 1313, 1473, 1647, 1845, 2045, 2272, 2521, 2785, 3076, 3398, 3744, 4115, 4548, 5010, 5524, 6086

Offset: 0

Gus Wiseman, Oct 06 2021

nonn

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.

			Partitions representing each of the a(7) = 11 reverse-alternating products:
     (7) -> 7
    (61) -> 1/6
    (52) -> 2/5
   (511) -> 5
    (43) -> 3/4
   (421) -> 2
  (4111) -> 1/4
   (331) -> 1
   (322) -> 3
  (3211) -> 2/3
  (2221) -> 1/2

The version for non-reverse alternating sum instead of product is A004526.
Counting only integers gives A028310, non-reverse A347707.
The version for factorizations is A038548, non-reverse A347460.
The non-reverse version is A347461.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A122768 counts distinct submultisets of partitions.
A126796 counts complete partitions.
A293627 counts knapsack factorizations by sum.
A301957 counts distinct subset-products of prime indices.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
A304793 counts distinct positive subset-sums of prime indices.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A001055, A002033, A002219, A028983, A119620, A325768, A345926, A347443, A347444, A347445, A347446.

revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[revaltprod/@IntegerPartitions[n]]],{n,0,30}]

cp's OEIS Frontend

A347437 Number of factorizations of n with integer alternating product.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A347438 Number of unordered factorizations of n with alternating product 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A347439 Number of factorizations of n with integer reciprocal alternating product.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A347442 Number of factorizations of n with integer reverse-alternating product.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A347463 Number of ordered factorizations of n with integer alternating product.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A347440 Number of factorizations of n with alternating product < 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A273013 Number of different arrangements of nonnegative integers on a pair of n-sided dice such that the dice can add to any integer from 0 to n^2-1.

Views