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A347464 Number of even-length ordered factorizations of n^2 into factors > 1 with alternating product 1. 9
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 22 19:45 EST 2022. Contains 350504 sequences. (Running on oeis4.)