A347438 Number of unordered factorizations of n with alternating product 1.

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

1,16

Comments

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)).

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

PlanetMath, alternating sum

Formula

a(2^n) = A035363(n).

a(n^2) = A001055(n).

Example

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

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[facs[n], altprod[#]==1&]], {n, 100}]

Prog

(PARI) 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

Crossrefs

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.

Sequence in context: A004530 A368980 A355549 * A374017 A373974 A245196

Adjacent sequences: A347435 A347436 A347437 * A347439 A347440 A347441

Keyword

nonn

Author

Gus Wiseman, Sep 06 2021

Extensions

Name and comments clarified (with unordered) by Jacob Sprittulla, Oct 05 2021

Status

approved