A347456 -id:A347456 - cp's OEIS frontend

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

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

A347450 Numbers whose multiset of prime indices has alternating product <= 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 14, 15, 16, 18, 21, 22, 24, 25, 26, 32, 33, 34, 35, 36, 38, 39, 40, 46, 49, 50, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 69, 72, 74, 77, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 98, 100, 104, 106, 111, 115, 118, 119, 121, 122

Offset: 1

Gus Wiseman, Sep 24 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also Heinz numbers integer partitions with reverse-alternating product <= 1, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers whose multiset of prime indices has alternating sum <= 1.

			The initial terms and their prime indices:
      1: {}            26: {1,6}           56: {1,1,1,4}
      2: {1}           32: {1,1,1,1,1}     57: {2,8}
      4: {1,1}         33: {2,5}           58: {1,10}
      6: {1,2}         34: {1,7}           60: {1,1,2,3}
      8: {1,1,1}       35: {3,4}           62: {1,11}
      9: {2,2}         36: {1,1,2,2}       64: {1,1,1,1,1,1}
     10: {1,3}         38: {1,8}           65: {3,6}
     14: {1,4}         39: {2,6}           69: {2,9}
     15: {2,3}         40: {1,1,1,3}       72: {1,1,1,2,2}
     16: {1,1,1,1}     46: {1,9}           74: {1,12}
     18: {1,2,2}       49: {4,4}           77: {4,5}
     21: {2,4}         50: {1,3,3}         81: {2,2,2,2}
     22: {1,5}         51: {2,7}           82: {1,13}
     24: {1,1,1,2}     54: {1,2,2,2}       84: {1,1,2,4}
     25: {3,3}         55: {3,5}           85: {3,7}

The additive version (alternating sum <= 0) is A028260.
The reverse version is A028982, counted by A119620.
Allowing any alternating product < 1 gives A119899.
Factorizations of this type are counted by A339846, complement A339890.
Allowing any alternating product >= 1 gives A344609, multiplicative A347456.
Partitions of this type are counted by A347443.
Allowing any integer alternating product gives A347454, reciprocal A347451.
The complement is A347465, reverse A028983, counted by A347448.
A056239 adds up prime indices, row sums of A112798.
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A344606 counts alternating permutations of prime indices.
A347457 lists Heinz numbers of partitions with integer alternating product.
Cf. A001105, A001222, A027193, A344617, A345958, A346703, A346704, A347449, A347461, A347462.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],altprod[primeMS[#]]<=1&]

Union of A028982 and A119899.

Union of A028260 and A001105.

A347454 Numbers whose multiset of prime indices has integer alternating product.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 108, 109, 112, 113

Offset: 1

Gus Wiseman, Sep 26 2021

nonn

First differs from A265640 in having 42.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also Heinz numbers of partitions with integer reverse-alternating product, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The terms and their prime indices begin:
      1: {}            20: {1,1,3}         47: {15}
      2: {1}           23: {9}             48: {1,1,1,1,2}
      3: {2}           25: {3,3}           49: {4,4}
      4: {1,1}         27: {2,2,2}         50: {1,3,3}
      5: {3}           28: {1,1,4}         52: {1,1,6}
      7: {4}           29: {10}            53: {16}
      8: {1,1,1}       31: {11}            59: {17}
      9: {2,2}         32: {1,1,1,1,1}     61: {18}
     11: {5}           36: {1,1,2,2}       63: {2,2,4}
     12: {1,1,2}       37: {12}            64: {1,1,1,1,1,1}
     13: {6}           41: {13}            67: {19}
     16: {1,1,1,1}     42: {1,2,4}         68: {1,1,7}
     17: {7}           43: {14}            71: {20}
     18: {1,2,2}       44: {1,1,5}         72: {1,1,1,2,2}
     19: {8}           45: {2,2,3}         73: {21}

The even-length case is A000290.
The additive version is A026424.
Allowing any alternating product < 1 gives A119899, strict A028260.
Allowing any alternating product >= 1 gives A344609, multiplicative A347456.
Factorizations of this type are counted by A347437.
These partitions are counted by A347445, reverse A347446.
Allowing any alternating product <= 1 gives A347450.
The reciprocal version is A347451.
The odd-length case is A347453.
The version for reversed prime indices is A347457, complement A347455.
Allowing any alternating product > 1 gives A347465, reverse A028983.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A344606 counts alternating permutations of prime indices.
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
Cf. A001105, A001222, A028982, A119620, A236913, A316523, A344653, A346703, A346704, A347443, A347439.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],IntegerQ[altprod[primeMS[#]]]&]

A347448 Number of integer partitions of n with alternating product > 1.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 12, 17, 25, 35, 49, 66, 90, 120, 161, 209, 275, 355, 460, 585, 750, 946, 1199, 1498, 1881, 2335, 2909, 3583, 4430, 5428, 6666, 8118, 9912, 12013, 14586, 17592, 21252, 25525, 30695, 36711, 43956, 52382, 62469, 74173, 88132, 104303, 123499

Offset: 0

Gus Wiseman, Sep 16 2021

nonn

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

			The a(2) = 1 through a(7) = 12 partitions:
  (2)  (3)   (4)    (5)     (6)      (7)
       (21)  (31)   (32)    (42)     (43)
             (211)  (41)    (51)     (52)
                    (311)   (222)    (61)
                    (2111)  (321)    (322)
                            (411)    (421)
                            (3111)   (511)
                            (21111)  (2221)
                                     (3211)
                                     (4111)
                                     (31111)
                                     (211111)

The strict case is A000009, except that a(0) = a(1) = 0.
Allowing any alternating product >= 1 gives A000041, reverse A344607.
Ranked by A028983 (reverse A347465), which has complement A028982.
The complement is counted by A119620, reverse A347443.
The multiplicative version is A339890, weak A347456, reverse A347705.
The even-length case is A344608.
Allowing any integer reverse-alternating product gives A347445.
Allowing any integer alternating product gives A347446.
The reverse version is A347449, also the odd-length case.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347461 counts possible alternating products of partitions.
Cf. A000070, A086543, A100824, A236913, A325534, A325535, A339846, A344654, A345196, A347440, A347444, A347462.

a:= n-> (p-> p(n)-p(iquo(n, 2)))(combinat[numbpart]):
seq(a(n), n=0..63);  # Alois P. Heinz, Oct 04 2021

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],altprod[#]>1&]],{n,0,30}]

a(n) = A000041(n) - A119620(n).

A347444 Number of odd-length integer partitions of n with integer alternating product.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 4, 8, 7, 14, 13, 24, 21, 40, 35, 62, 55, 99, 85, 151, 128, 224, 195, 331, 283, 481, 416, 690, 593, 980, 844, 1379, 1189, 1918, 1665, 2643, 2292, 3630, 3161, 4920, 4299, 6659, 5833, 8931, 7851, 11905, 10526, 15805, 13987, 20872, 18560, 27398

Offset: 0

Gus Wiseman, Sep 14 2021

nonn

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

The reverse version (integer reverse-alternating product) is the same.

			The a(1) = 1 through a(9) = 14 partitions:
  (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)        (9)
            (111)  (211)  (221)    (222)    (322)      (332)      (333)
                          (311)    (411)    (331)      (422)      (441)
                          (11111)  (21111)  (421)      (611)      (522)
                                            (511)      (22211)    (621)
                                            (22111)    (41111)    (711)
                                            (31111)    (2111111)  (22221)
                                            (1111111)             (32211)
                                                                  (33111)
                                                                  (42111)
                                                                  (51111)
                                                                  (2211111)
                                                                  (3111111)
                                                                  (111111111)

The reciprocal version is A035363.
Allowing any alternating product gives A027193.
The multiplicative version (factorizations) is A347441.
Allowing any length gives A347446, reverse A347445.
Allowing any length and alternating product > 1 gives A347448.
Allowing any reverse-alternating product > 1 gives A347449.
Ranked by A347453.
The even-length instead of odd-length version is A347704.
A000041 counts partitions.
A000302 counts odd-length compositions, ranked by A053738.
A025047 counts wiggly compositions.
A026424 lists numbers with odd bigomega.
A027187 counts partitions of even length, strict A067661.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A339890 counts odd-length factorizations.
A347437 counts factorizations with integer alternating product.
A347461 counts possible alternating products of partitions.
Cf. A000070, A236559, A236913, A236914, A304620, A344654, A347439, A347442, A347456, A347457, A347460, A347462, A347463.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]

A347458 Number of factorizations of n^2 with integer alternating product.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 17, 2, 6, 6, 15, 2, 17, 2, 16, 6, 6, 2, 41, 4, 6, 8, 16, 2, 31, 2, 27, 6, 6, 6, 56, 2, 6, 6, 39, 2, 31, 2, 17, 17, 6, 2, 90, 4, 17, 6, 17, 2, 41, 6, 39, 6, 6, 2, 105, 2, 6, 17, 48, 6, 31, 2, 17, 6, 31, 2, 148, 2, 6, 17, 17, 6, 32, 2, 86, 15, 6, 2, 107, 6, 6, 6, 40, 2, 109, 6, 17

Offset: 1

Gus Wiseman, Sep 21 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.
The even-length case, the case of alternating product 1, and the case of alternating sum 0 are all counted by A001055.

			The a(2) = 2 through a(8) = 8 factorizations:
  4     9     16        25    36        49    64
  2*2   3*3   4*4       5*5   6*6       7*7   8*8
              2*2*4           2*2*9           2*4*8
              2*2*2*2         2*3*6           4*4*4
                              3*3*4           2*2*16
                              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..16415

Positions of 2's are A000040, squares A001248.
The restriction to powers of 2 is A344611.
This is the restriction to perfect squares of A347437.
The nonsquared even-length version is A347438.
The reciprocal version is A347459, non-squared A347439.
The additive version (partitions) is the even bisection of A347446.
The nonsquared ordered version is A347463.
The case of alternating product 1 in the ordered version is A347464.
Allowing any alternating product gives A347466.
A000290 lists squares, complement A000037.
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.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347457 ranks partitions with integer alternating product.
Cf. A062312, A119620, A330972, A346635, A347440, A347441, A347442, A347445, A347451, A347456, A347704, A347705.
Apparently, A006881 gives the positions of 6's. - Antti Karttunen, Oct 22 2023

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^2],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))));
A347458(n) = A347437(n*n); \\ Antti Karttunen, Oct 22 2023

a(2^n) = A344611(n).

a(n) = A347437(n^2).

Data section extended up to a(92) by Antti Karttunen, Oct 22 2023

A347449 Number of integer partitions of n with reverse-alternating product > 1.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 5, 10, 11, 20, 22, 37, 41, 66, 75, 113, 129, 190, 218, 310, 358, 497, 576, 782, 908, 1212, 1411, 1851, 2156, 2793, 3255, 4163, 4853, 6142, 7159, 8972, 10451, 12989, 15123, 18646, 21689, 26561, 30867, 37556, 43599, 52743, 61161, 73593

Offset: 0

Gus Wiseman, Sep 16 2021

nonn

All such partitions have odd length.

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(2) = 1 through a(9) = 11 partitions:
  (2)  (3)  (4)    (5)    (6)      (7)      (8)        (9)
            (211)  (311)  (222)    (322)    (332)      (333)
                          (321)    (421)    (422)      (432)
                          (411)    (511)    (431)      (522)
                          (21111)  (31111)  (521)      (531)
                                            (611)      (621)
                                            (22211)    (711)
                                            (32111)    (32211)
                                            (41111)    (42111)
                                            (2111111)  (51111)
                                                       (3111111)

The strict case is A067659, except that a(0) = a(1) = 0.
The even bisection is A236559.
The non-reverse multiplicative version is A339890, weak A347456.
The case of >= 1 instead of > 1 is A344607.
The opposite version is A344608, also the non-reverse even-length case.
The complement is counted by A347443, non-reverse A119620.
Allowing any integer reverse-alternating product gives A347445.
Allowing any integer alternating product gives A347446.
Reverse version of A347448; also the odd-length case.
The Heinz numbers of these partitions are the complement of A347450.
The multiplicative version (factorizations) is A347705.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A100824 counts partitions of n with alternating sum <= 1.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347462 counts possible reverse-alternating products of partitions.
Cf. A000070, A008549, A086543, A182616, A236913, A325534, A325535, A344611, A347442, A347444, A347447, A347453, A347461, A347465.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],altprod[Reverse[#]]>1&]],{n,0,30}]

a(n) = A344607(n) - A119620(n).

A347465 Numbers whose multiset of prime indices has alternating product > 1.

Original entry on oeis.org

3, 5, 7, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 30, 31, 37, 41, 42, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 66, 67, 68, 70, 71, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 112, 113, 114, 116, 117, 120, 124, 125, 127

Offset: 1

Gus Wiseman, Sep 27 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
All terms have odd bigomega (A001222).
Also Heinz numbers integer partitions with reverse-alternating product > 1.

			The terms and their prime indices begin:
      3: {2}         37: {12}            68: {1,1,7}
      5: {3}         41: {13}            70: {1,3,4}
      7: {4}         42: {1,2,4}         71: {20}
     11: {5}         43: {14}            73: {21}
     12: {1,1,2}     44: {1,1,5}         75: {2,3,3}
     13: {6}         45: {2,2,3}         76: {1,1,8}
     17: {7}         47: {15}            78: {1,2,6}
     19: {8}         48: {1,1,1,1,2}     79: {22}
     20: {1,1,3}     52: {1,1,6}         80: {1,1,1,1,3}
     23: {9}         53: {16}            83: {23}
     27: {2,2,2}     59: {17}            89: {24}
     28: {1,1,4}     61: {18}            92: {1,1,9}
     29: {10}        63: {2,2,4}         97: {25}
     30: {1,2,3}     66: {1,2,5}         99: {2,2,5}
     31: {11}        67: {19}           101: {26}

The squarefree case is A030059 without 2.
The reverse version is A028983, counted by A119620.
The opposite version (< 1 instead of > 1) is A119899.
Factorizations of this type are counted by A339890, reverse A347705.
The weak version (>= 1 instead of > 1) is A344609.
Partitions of this type are counted by A347449, reverse A347448.
The complement is A347450, counted by A339846 or A347443.
Allowing any integer reverse-alternating product gives A347454.
Allowing any integer alternating product gives A347457.
A335433 ranks inseparable partitions, complement A335448.
A347446 counts partitions with integer alternating product, reverse A347445.
Cf. A008549, A344607, A344608, A344611, A347442, A347444, A347447, A347453, A347456, A347461, A347462.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],altprod[primeMS[#]]>1&]

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

cp's OEIS Frontend

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

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A347450 Numbers whose multiset of prime indices has alternating product <= 1.

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A347454 Numbers whose multiset of prime indices has integer alternating product.

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A347448 Number of integer partitions of n with alternating product > 1.

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A347444 Number of odd-length integer partitions of n with integer alternating product.

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A347458 Number of factorizations of n^2 with integer alternating product.

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A347449 Number of integer partitions of n with reverse-alternating product > 1.

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A347465 Numbers whose multiset of prime indices has alternating product > 1.

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