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

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

Offset

1,16

Comments

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).

Links

Table of n, a(n) for n=1..87.

Formula

a(2^n) = A027187(n).

a(n^2) = A347459(n).

Example

The a(n) factorizations for n = 16, 36, 64, 72, 128, 144:
  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

Mathematica

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}]

Crossrefs

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.

Sequence in context: A318499 A346012 A317946 * A347048 A140807 A232629

Adjacent sequences: A347436 A347437 A347438 * A347440 A347441 A347442

Keyword

nonn

Author

Gus Wiseman, Sep 07 2021

Status

approved