A347463 Number of ordered factorizations of n with integer alternating product.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 7, 1, 4, 1, 4, 1, 1, 1, 6, 2, 1, 3, 4, 1, 1, 1, 11, 1, 1, 1, 18, 1, 1, 1, 6, 1, 1, 1, 4, 4, 1, 1, 20, 2, 4, 1, 4, 1, 6, 1, 6, 1, 1, 1, 8, 1, 1, 4, 26, 1, 1, 1, 4, 1, 1, 1, 35, 1, 1, 4, 4, 1, 1, 1, 20, 7, 1, 1, 8, 1, 1, 1, 6, 1, 8, 1, 4, 1, 1, 1, 32, 1, 4, 4, 18

Offset: 1

Author

Gus Wiseman, Oct 07 2021

Keywords

• nonn

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)).

Examples

			The ordered factorizations for n = 4, 8, 12, 16, 24, 32, 36:
  4     8       12      16        24      32          36
  2*2   4*2     6*2     4*4       12*2    8*4         6*6
        2*2*2   2*2*3   8*2       2*2*6   16*2        12*3
                3*2*2   2*2*4     3*2*4   2*2*8       18*2
                        2*4*2     4*2*3   2*4*4       2*2*9
                        4*2*2     6*2*2   4*2*4       2*3*6
                        2*2*2*2           4*4*2       2*6*3
                                          8*2*2       3*2*6
                                          2*2*4*2     3*3*4
                                          4*2*2*2     3*6*2
                                          2*2*2*2*2   4*3*3
                                                      6*2*3
                                                      6*3*2
                                                      9*2*2
                                                      2*2*3*3
                                                      2*3*3*2
                                                      3*2*2*3
                                                      3*3*2*2
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Positions of 2's are A001248.

Positions of 1's are A005117.

The restriction to powers of 2 is A116406.

The even-length case is A347048

The odd-length case is A347049.

The unordered version is A347437, reciprocal A347439, reverse A347442.

The case of partitions is A347446, reverse A347445, ranked by A347457.

A001055 counts factorizations (strict A045778, ordered A074206).

A046099 counts factorizations with no alternating permutations.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A119620 counts partitions with alternating product 1, ranked by A028982.

A273013 counts ordered factorizations of n^2 with alternating product 1.

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.

Cf. A025047, A038548, A138364, A347440, A347441, A347453, A347454, A347456, A347458, A347459, A347464, A347705, A347708.

Programs

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],IntegerQ[altprod[#]]&]],{n,100}]

PARI

A347463(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, A347463(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

Formula

a(n) = A347048(n) + A347049(n).

Extensions

Data section extended up to a(100) by Antti Karttunen, Jul 28 2024