A062312 -id:A062312 - 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

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

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

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

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

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

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)).

A275647 Decimal expansion of Pi^2/6 - Sum_{k>=1} 1/prime(k)^2.

Original entry on oeis.org

1, 1, 9, 2, 6, 8, 6, 6, 4, 6, 8, 0, 7, 1, 6, 0, 9, 3, 7, 9, 6, 5, 8, 7, 1, 8, 0, 1, 8, 1, 3, 7, 7, 7, 2, 5, 5, 0, 4, 5, 7, 1, 8, 5, 5, 7, 9, 6, 6, 9, 0, 6, 0, 1, 5, 9, 9, 9, 1, 3, 9, 2, 9, 9, 0, 1, 8, 8, 4, 2, 4, 4, 3, 0, 3, 9, 2, 8, 9, 9, 9, 9, 3, 8, 4, 6, 7, 1, 3, 5, 6, 2, 6, 4, 8, 5, 1, 8, 4, 7, 0, 8, 1, 1, 9, 8, 0, 9, 1, 2, 9, 5, 4, 1, 6, 5, 5, 9, 6, 6

Offset: 1

Ilya Gutkovskiy, Aug 04 2016

Decimal expansion of sum of squares of reciprocals of nonprime numbers.
Decimal expansion of the nonprime zeta function at 2.
Continued fraction [1; 5, 5, 3, 1, 2, 2, 6, 2, 2, 4, 1, 1, 93, 2, 1, 1, 5, 3, 5, 3, 2, 1, 2, 6, 1, 4, 5, 1, 34, 1, ...]
More generally, the nonprime zeta function at s equals Sum_{k>=1} (1/k^s - 1/prime(k)^s) = Product_{k>=1} 1/(1 - prime(k)^(-s)) - Sum_{k>=1} 1/prime(k)^s.
Floor(1/(zeta(s)-prime zeta(s)-1)) gives second term in continued fraction for nonprime zeta(s): 5, 36, 187, 852, 3663, 15280, 62692, 254760, 1029279, 4143617, ...
Dirichlet g.f. of A005171: nonprime zeta(s).

			1/1^2 + 1/4^2 + 1/6^2 + 1/8^2 + 1/9^2 + 1/10^2 + ... = 1.192686646807160937965871801813777255045718557966906015999...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Ilya Gutkovskiy, Nonprime zeta function.
Eric Weisstein's World of Mathematics, Riemann Zeta Function 2.
Eric Weisstein's World of Mathematics, Prime Zeta Function.

Cf. A002808, A008683, A013661, A018252, A062312, A085548, A111003, A222171.

RealDigits[Pi^2/6 - PrimeZetaP[2], 10, 120][[1]]
RealDigits[Zeta[2] - PrimeZetaP[2], 10, 120][[1]]

eps()=2.>>bitprecision(1.)
primezeta(s)=my(lm=s*log(2)); lm=lambertw(lm/eps())\lm; sum(k=1,lm, moebius(k)*log(abs(zeta(k*s)))/k)
zeta(2) - primezeta(2) \\ Charles R Greathouse IV, Aug 05 2016

Pi^2/6 - sumeulerrat(1/p, 2) \\ Amiram Eldar, Mar 19 2021

Equals zeta(2) - prime zeta(2) = A013661 - A085548.
Equals Sum_{k>=1} (1 - k*mu(k)*log(zeta(2*k)))/k^2, where mu(k) is the Moebius function (A008683).
Equals Sum_{k>=1} 1/A062312(k).
Equals Sum_{k>=1} 1/A018252(k)^2.
Equals 1 + Sum_{k>=1} 1/A002808(k)^2.
Equals A222171 + A111003 - A085548.

A374370 Square array read by antidiagonals: the n-th row lists n-gonal numbers that are products of smaller n-gonal numbers.

Original entry on oeis.org

1, 4, 1, 6, 36, 1, 8, 45, 16, 1, 9, 210, 36, 10045, 1, 10, 300, 64, 11310, 2850, 1, 12, 378, 81, 20475, 61776, 6426, 1, 14, 630, 100, 52360, 79800, 9828, 1408, 1, 15, 780, 144, 197472, 103740, 35224, 61920, 265926, 1, 16, 990, 196, 230300, 145530, 60606, 67200, 391950, 69300, 1

Offset: 2

Pontus von Brömssen, Jul 07 2024

If there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.

The first term in each row is 1, because 1 is an n-gonal number for every n and it equals the empty product.

			Array begins:
   n=2: 1,      4,       6,       8,       9,       10,       12,       14
   n=3: 1,     36,      45,     210,     300,      378,      630,      780
   n=4: 1,     16,      36,      64,      81,      100,      144,      196
   n=5: 1,  10045,   11310,   20475,   52360,   197472,   230300,   341055
   n=6: 1,   2850,   61776,   79800,  103740,   145530,   437580,   719400
   n=7: 1,   6426,    9828,   35224,   60606,  1349460,  2077992,  3333330
   n=8: 1,   1408,   61920,   67200,  276640,   297045,   870485,  1022000
   n=9: 1, 265926,  391950, 1096200, 1767546,  1787500,  9909504, 28123200
  n=10: 1,  69300, 1297890, 4257000, 5756400,  9140040,  9729720, 10648800
  n=11: 1,  79135,  792330, 2382380, 5570565, 15361500, 22230000, 49888395
  n=12: 1,   9504,   45696,  604128, 1981980,  2208465,  4798080, 13837824

Cf. A057145, A374371 (second column), A374498.

Rows: A018252 (n=2), A068143 (n=3 except first term), A062312 (n=4), A374372 (n=5), A374373 (n=6).

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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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.

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A347441 Number of odd-length factorizations 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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A347459 Number of factorizations of n^2 with integer reciprocal alternating product.

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