%I #27 Dec 04 2021 13:03:22

%S 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,

%T 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,

%U 0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0

%N Number of unordered factorizations of n with alternating product 1.

%C Also the number of unordered factorizations of n with alternating sum 0.

%C Also the number of unordered factorizations of n with all even multiplicities.

%C This is the even-length case of A347437, the odd-length case being A347441.

%C An unordered factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

%C We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

%H Antti Karttunen, <a href="/A347438/b347438.txt">Table of n, a(n) for n = 1..65537</a>

%H PlanetMath, <a href="https://planetmath.org/alternatingsum">alternating sum</a>

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

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

%e The a(n) factorizations for n = 16, 64, 144, 256, 576:

%e 4*4 8*8 12*12 16*16 24*24

%e 2*2*2*2 2*2*4*4 2*2*6*6 2*2*8*8 3*3*8*8

%e 2*2*2*2*2*2 3*3*4*4 4*4*4*4 4*4*6*6

%e 2*2*2*2*3*3 2*2*2*2*4*4 2*2*12*12

%e 2*2*2*2*2*2*2*2 2*2*2*2*6*6

%e 2*2*3*3*4*4

%e 2*2*2*2*2*2*3*3

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];

%t Table[Length[Select[facs[n],altprod[#]==1&]],{n,100}]

%o (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

%Y Positions of zeros are A000037.

%Y Positions of nonzero terms are A000290.

%Y The restriction to perfect squares is A001055 (ordered: A273013).

%Y The restriction to powers of 2 is A035363.

%Y The additive version is A119620, ranked by A028982.

%Y Positions of non-1's are A213367 \ {1}.

%Y Positions of 1's are A280076 = {1} \/ A001248.

%Y Sorted first positions are 1, 2, and all terms of A330972 squared.

%Y Allowing any alternating product <= 1 gives A339846.

%Y Allowing any alternating product > 1 gives A339890.

%Y Allowing any integer alternating product gives A347437.

%Y Allowing any integer reciprocal alternating product gives A347439.

%Y Allowing any alternating product < 1 gives A347440.

%Y Allowing any alternating product >= 1 gives A347456.

%Y A046099 counts factorizations with no alternating permutations.

%Y A071321 gives the alternating sum of prime factors (reverse: A071322).

%Y A316524 gives the alternating sum of prime indices (reverse: A344616).

%Y A344606 counts alternating permutations of prime factors.

%Y A347441 counts odd-length factorizations with integer alternating product.

%Y A347460 counts possible alternating products of factorizations.

%Y Cf. A000041, A005117, A025047, A038548, A062312, A088218, A316523, A332269, A344607, A347442, A347446, A347463.

%K nonn

%O 1,16

%A _Gus Wiseman_, Sep 06 2021

%E Name and comments clarified (with unordered) by _Jacob Sprittulla_, Oct 05 2021