A347463 -id:A347463 - 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

A347446 Number of integer partitions of n with integer alternating product.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 31, 37, 54, 62, 84, 100, 134, 157, 207, 241, 314, 363, 463, 537, 685, 785, 985, 1138, 1410, 1616, 1996, 2286, 2801, 3201, 3885, 4434, 5363, 6098, 7323, 8329, 9954, 11293, 13430, 15214, 18022, 20383, 24017, 27141, 31893, 35960

Offset: 0

Gus Wiseman, Sep 15 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(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (41)     (33)      (61)
             (111)  (31)    (221)    (42)      (322)
                    (211)   (311)    (51)      (331)
                    (1111)  (2111)   (222)     (421)
                            (11111)  (411)     (511)
                                     (2211)    (2221)
                                     (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

Allowing any reverse-alternating product >= 1 gives A344607.
Allowing any alternating product <= 1 gives A119620, reverse A347443.
Allowing any reverse-alternating product < 1 gives A344608.
The multiplicative version (factorizations) is A347437, reverse A347442.
The odd-length case is A347444, ranked by A347453.
The reverse version is A347445, ranked by A347454.
Allowing any alternating product > 1 gives A347448, reverse A347449.
Ranked by A347457.
The even-length case is A347704.
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).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A347461 counts possible alternating products of partitions.
Cf. A025047, A067661, A339890, A347450, A344654, A344740, A347439, A347440, A347451, A347463, A347705.

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

A347456 Number of factorizations of n with alternating product >= 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 09 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.
Also the number of factorizations of n with alternating sum >= 0.

			The a(n) factorizations for n = 4, 16, 24, 36, 60, 64, 96:
  4     16        24      36        60       64            96
  2*2   4*4       2*2*6   6*6       2*5*6    8*8           2*6*8
        2*2*4     2*3*4   2*2*9     3*4*5    2*4*8         3*4*8
        2*2*2*2           2*3*6     2*2*15   4*4*4         4*4*6
                          3*3*4     2*3*10   2*2*16        2*2*24
                          2*2*3*3            2*2*4*4       2*3*16
                                             2*2*2*2*4     2*4*12
                                             2*2*2*2*2*2   2*2*2*2*6
                                                           2*2*2*3*4

The case of partitions is A000041, reverse A344607.
The reverse version is A001055, strict A347705.
Positions of 3's appear to be A065036.
Positions of 1's are 1 and A167171.
The opposite version (<= instead of >=) is A339846.
The strict version (> instead of >=) is A339890, also the odd-length case.
Allowing any integer alternating product gives A347437.
The case of alternating product 1 is A347438, also the even-length case.
Allowing any integer reciprocal alternating product gives A347439.
The complement (< instead of >=) is A347440.
Allowing any integer reverse-alternating product gives A347442.
A038548 counts factorizations with a wiggly permutation.
A045778 counts strict factorizations.
A074206 counts ordered factorizations.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1.
A347447 counts strict factorizations with alternating product > 1.
Cf. A001700, A028983, A316523, A347441, A347443, A347446, A347448, A347450, A347454, A347463, 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[facs[n],altprod[#]>=1&]],{n,100}]

a(n) = A347438(n) + A347440(n).

A348379 Number of factorizations of n with an alternating permutation.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 1, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 1, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 3, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 28 2021

nonn

First differs from A335434 at a(216) = 27, A335434(216) = 28. Also differs from A335434 at a(270) = 19, A335434(270) = 20.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
All of the counted factorizations are separable (A335434).
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 {1..n}.

			The a(270) = 19 factorizations:
  (2*3*3*15)  (2*3*45)  (2*135)  (270)
  (2*3*5*9)   (2*5*27)  (3*90)
  (3*3*5*6)   (2*9*15)  (5*54)
              (3*3*30)  (6*45)
              (3*5*18)  (9*30)
              (3*6*15)  (10*27)
              (3*9*10)  (15*18)
              (5*6*9)

Wikipedia, Alternating permutation

Partitions not of this type are counted by A345165, ranked by A345171.
Partitions of this type are counted by A345170, ranked by A345172.
Twins and partitions of this type are counted by A344740, ranked by A344742.
The case with twins is A347050.
The complement is counted by A348380, without twins A347706.
The ordered version is A348610.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A038548, A056986, A325534, A335452, A347437, A347438, A347439, A347442, A347456, A347463, A348381, A348383, A348609.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[facs[n],Select[Permutations[#],wigQ]!={}&]],{n,100}]

a(2^n) = A345170(n).

A348610 Number of alternating ordered factorizations of n.

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, 12, 1, 3, 3, 6, 1, 11, 1, 7, 3, 3, 3, 15, 1, 3, 3, 12, 1, 11, 1, 6, 6, 3, 1, 23, 1, 6, 3, 6, 1, 12, 3, 12, 3, 3, 1, 28, 1, 3, 6, 12, 3, 11, 1, 6, 3, 11, 1, 33, 1, 3, 6, 6, 3, 11, 1, 23, 4, 3

Offset: 1

Gus Wiseman, Nov 05 2021

nonn

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

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).

			The alternating ordered factorizations of n = 1, 6, 12, 16, 24, 30, 32, 36:
  ()   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*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                   3*6*2
                                                     6*2*3
                                                     2*3*2*3
                                                     3*2*3*2

Wikipedia, Alternating permutation

The additive version (compositions) is A025047 ranked by A345167.
The complementary additive version is A345192, ranked by A345168.
Dominated by A348611 (the anti-run version) at positions A122181.
The complement is counted by A348613.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A347463 counts ordered factorizations with integer alternating product.
A348379 counts factorizations w/ an alternating permutation.
A348380 counts factorizations w/o an alternating permutation.
Cf. A038548, A049774, A056986, A102726, A128761, A344614, A347050, A347464, A347706, A348383.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]] == Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[ordfacs[n],wigQ]],{n,100}]

A347706 Number of factorizations of n that are not a twin (x*x) nor have an alternating permutation.

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 28 2021

nonn

First differs from A348381 at a(216) = 4, A348381(216) = 3.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
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.

			The a(n) factorizations for n = 96, 192, 2160, 576:
  2*2*2*12      3*4*4*4         3*3*3*80       4*4*4*9
  2*2*2*2*6     2*2*2*24        6*6*6*10       2*2*2*72
  2*2*2*2*2*3   2*2*2*2*12      2*2*2*270      2*2*2*2*36
                2*2*2*2*2*6     2*3*3*3*40     2*2*2*2*4*9
                2*2*2*2*3*4     2*2*2*2*135    2*2*2*2*6*6
                2*2*2*2*2*2*3   2*2*2*2*3*45   2*2*2*2*2*18
                                2*2*2*2*5*27   2*2*2*2*3*12
                                2*2*2*2*9*15   2*2*2*2*2*2*9
                                               2*2*2*2*2*3*6
                                               2*2*2*2*2*2*3*3

Wikipedia, Alternating permutation

Positions of nonzero terms are A046099.
Partitions of this type are counted by A344654, ranked by A344653.
Partitions not of this type are counted by A344740, ranked by A344742.
The complement is counted by A347050, without twins A348379.
The version for compositions is A348377.
The version allowing twins is A348380.
The inseparable case is A348381.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations of sets.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347438 counts factorizations with alternating product 1, additive A119620.
A348610 counts alternating ordered factorizations.
Cf. A049774, A325535, A336103, A344614, A347437, A347439, A347442, A347456, A347463, A348382, A348383.

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],Function[f,Select[Permutations[f],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]=={}]]],{n,100}]

a(2^n) = A344654(n).

A348613 Number of non-alternating ordered factorizations of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 8, 1, 0, 1, 2, 0, 2, 0, 9, 0, 0, 0, 11, 0, 0, 0, 8, 0, 2, 0, 2, 2, 0, 0, 25, 1, 2, 0, 2, 0, 8, 0, 8, 0, 0, 0, 16, 0, 0, 2, 20, 0, 2, 0, 2, 0, 2, 0, 43, 0, 0, 2, 2, 0, 2, 0, 25, 4, 0, 0, 16, 0

Offset: 1

Gus Wiseman, Nov 03 2021

nonn

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

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.

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

Wikipedia, Alternating permutation

The complementary additive version is A025047, ranked by A345167.
The additive version is A345192, ranked by A345168, without twins A348377.
The complement is counted by A348610.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions without an alternating permutation, ranked by A345171.
A345170 counts partitions with an alternating permutation, ranked by A345172.
A348379 counts factorizations w/ an alternating permutation, with twins A347050.
A348380 counts factorizations w/o an alternating permutation, w/o twins A347706.
A348611 counts anti-run ordered factorizations.
Cf. A038548, A056986, A344614, A344653, A344654, A344740, A347437, A347438, A347463, A348381, A348609.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[ordfacs[n],!wigQ[#]&]],{n,100}]

cp's OEIS Frontend

A347437 Number of factorizations of n with integer alternating product.

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

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

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

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A347446 Number of integer partitions of n with integer alternating product.

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A347456 Number of factorizations of n with alternating product >= 1.

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A348379 Number of factorizations of n with an alternating permutation.

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