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%I #18 Nov 03 2021 09:09:28
%S 1,1,1,2,1,5,1,6,2,5,1,26,1,5,5,20,1,26,1,26,5,5,1,134,2,5,6,26,1,73,
%T 1,70,5,5,5,230,1,5,5,134,1,73,1,26,26,5,1,670,2,26,5,26,1,134,5,134,
%U 5,5,1,686,1,5,26,252,5,73,1,26,5,73,1,1714,1,5,26
%N Number of even-length ordered factorizations of n^2 into factors > 1 with alternating product 1.
%C An ordered factorization of n is a 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)).
%C Also the number of ordered pairs of ordered factorizations of n, both of the same length.
%C Note that the version for all n (not just squares) is 0 except at perfect squares.
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%e The a(12) = 26 ordered factorizations:
%e (2*2*6*6) (3*2*4*6) (6*2*2*6) (4*2*3*6) (12*12)
%e (2*3*6*4) (3*3*4*4) (6*3*2*4) (4*3*3*4)
%e (2*4*6*3) (3*4*4*3) (6*4*2*3) (4*4*3*3)
%e (2*6*6*2) (3*6*4*2) (6*6*2*2) (4*6*3*2)
%e (2*2*2*2*3*3) (3*2*2*2*2*3)
%e (2*2*2*3*3*2) (3*2*2*3*2*2)
%e (2*2*3*2*2*3) (3*3*2*2*2*2)
%e (2*2*3*3*2*2)
%e (2*3*2*2*3*2)
%e (2*3*3*2*2*2)
%e For example, the ordered factorization 6*3*2*4 = 144 has alternating product 6/3*2/4 = 1, so is counted under a(12).
%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[Join@@Permutations/@facs[n^2],EvenQ[Length[#]]&&altprod[#]==1&]],{n,100}]
%o (PARI)
%o A347464aux(n, k=0, t=1) = if(1==n, (0==k)&&(1==t), my(s=0); fordiv(n, d, if((d>1), s += A347464aux(n/d, 1-k, t*(d^((-1)^k))))); (s));
%o A347464(n) = A347464aux(n^2); \\ _Antti Karttunen_, Oct 30 2021
%Y Positions of 1's are A008578 (1 and A000040).
%Y The restriction to powers of 2 is A000984.
%Y Positions of 2's are A001248.
%Y The not necessarily even-length version is A273013.
%Y A000290 lists squares, complement A000037.
%Y A001055 counts factorizations.
%Y A027187 counts even-length partitions.
%Y A074206 counts ordered factorizations.
%Y A119620 counts partitions with alternating product 1, ranked by A028982.
%Y A339846 counts even-length factorizations, ordered A347706.
%Y A347438 counts factorizations with alternating product 1.
%Y A347457 ranks partitions with integer alternating product.
%Y A347460 counts possible alternating products of factorizations.
%Y A347466 counts factorizations of n^2.
%Y Cf. A062312, A339890, A347437, A347439, A347440, A347442, A347456, A347459, A347463, A347705.
%K nonn
%O 1,4
%A _Gus Wiseman_, Sep 23 2021
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