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

Author

Gus Wiseman, Sep 06 2021

Keywords

• nonn

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)).

Examples

			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
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• PlanetMath, alternating sum

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.

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

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

Formula

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

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

Extensions

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