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

%I A347438 #27 Dec 04 2021 13:03:22
%S A347438 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 A347438 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 A347438 0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0
%N A347438 Number of unordered factorizations of n with alternating product 1.
%C A347438 Also the number of unordered factorizations of n with alternating sum 0.
%C A347438 Also the number of unordered factorizations of n with all even multiplicities.
%C A347438 This is the even-length case of A347437, the odd-length case being A347441.
%C A347438 An unordered factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A347438 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%H A347438 Antti Karttunen, <a href="/A347438/b347438.txt">Table of n, a(n) for n = 1..65537</a>
%H A347438 PlanetMath, <a href="https://planetmath.org/alternatingsum">alternating sum</a>
%F A347438 a(2^n) = A035363(n).
%F A347438 a(n^2) = A001055(n).
%e A347438 The a(n) factorizations for n = 16, 64, 144, 256, 576:
%e A347438   4*4      8*8          12*12        16*16            24*24
%e A347438   2*2*2*2  2*2*4*4      2*2*6*6      2*2*8*8          3*3*8*8
%e A347438            2*2*2*2*2*2  3*3*4*4      4*4*4*4          4*4*6*6
%e A347438                         2*2*2*2*3*3  2*2*2*2*4*4      2*2*12*12
%e A347438                                      2*2*2*2*2*2*2*2  2*2*2*2*6*6
%e A347438                                                       2*2*3*3*4*4
%e A347438                                                       2*2*2*2*2*2*3*3
%t A347438 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347438 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347438 Table[Length[Select[facs[n],altprod[#]==1&]],{n,100}]
%o A347438 (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 A347438 Positions of zeros are A000037.
%Y A347438 Positions of nonzero terms are A000290.
%Y A347438 The restriction to perfect squares is A001055 (ordered: A273013).
%Y A347438 The restriction to powers of 2 is A035363.
%Y A347438 The additive version is A119620, ranked by A028982.
%Y A347438 Positions of non-1's are A213367 \ {1}.
%Y A347438 Positions of 1's are A280076 = {1} \/ A001248.
%Y A347438 Sorted first positions are 1, 2, and all terms of A330972 squared.
%Y A347438 Allowing any alternating product <= 1 gives A339846.
%Y A347438 Allowing any alternating product > 1 gives A339890.
%Y A347438 Allowing any integer alternating product gives A347437.
%Y A347438 Allowing any integer reciprocal alternating product gives A347439.
%Y A347438 Allowing any alternating product < 1 gives A347440.
%Y A347438 Allowing any alternating product >= 1 gives A347456.
%Y A347438 A046099 counts factorizations with no alternating permutations.
%Y A347438 A071321 gives the alternating sum of prime factors (reverse: A071322).
%Y A347438 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A347438 A344606 counts alternating permutations of prime factors.
%Y A347438 A347441 counts odd-length factorizations with integer alternating product.
%Y A347438 A347460 counts possible alternating products of factorizations.
%Y A347438 Cf. A000041, A005117, A025047, A038548, A062312, A088218, A316523, A332269, A344607, A347442, A347446, A347463.
%K A347438 nonn
%O A347438 1,16
%A A347438 _Gus Wiseman_, Sep 06 2021
%E A347438 Name and comments clarified (with unordered) by _Jacob Sprittulla_, Oct 05 2021