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

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

1,4

Comments

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.

Links

Table of n, a(n) for n=1..75.

Index entries for sequences computed from exponents in factorization of n

Example

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).

Mathematica

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}]

Prog

(PARI)
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

Crossrefs

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.

Sequence in context: A090080 A151737 A211361 * A249548 A014650 A014648

Adjacent sequences: A347461 A347462 A347463 * A347465 A347466 A347467

Keyword

nonn

Author

Gus Wiseman, Sep 23 2021

Status

approved