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A347437 Number of factorizations of n with integer alternating product.

%I A347437 #16 Oct 22 2023 16:43:04
%S A347437 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,4,1,1,
%T A347437 1,6,1,1,1,2,1,1,1,2,2,1,1,5,2,2,1,2,1,3,1,2,1,1,1,2,1,1,2,8,1,1,1,2,
%U A347437 1,1,1,6,1,1,2,2,1,1,1,5,4,1,1,2,1,1,1,2,1,3,1,2,1,1,1,6,1,2,2,6,1,1,1,2,1,1,1,7
%N A347437 Number of factorizations of n with integer alternating product.
%C A347437 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A347437 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%H A347437 Antti Karttunen, <a href="/A347437/b347437.txt">Table of n, a(n) for n = 1..65537</a>
%H A347437 PlanetMath, <a href="https://planetmath.org/alternatingsum">alternating sum</a>
%F A347437 a(2^n) = A344607(n).
%F A347437 a(n^2) = A347458(n).
%e A347437 The factorizations for n = 4, 16, 36, 48, 54, 64, 108:
%e A347437   (4)   (16)      (36)      (48)        (54)    (64)          (108)
%e A347437   (2*2) (4*4)     (6*6)     (2*4*6)     (2*3*9) (8*8)         (2*6*9)
%e A347437         (2*2*4)   (2*2*9)   (3*4*4)     (3*3*6) (2*4*8)       (3*6*6)
%e A347437         (2*2*2*2) (2*3*6)   (2*2*12)            (4*4*4)       (2*2*27)
%e A347437                   (3*3*4)   (2*2*2*2*3)         (2*2*16)      (2*3*18)
%e A347437                   (2*2*3*3)                     (2*2*4*4)     (3*3*12)
%e A347437                                                 (2*2*2*2*4)   (2*2*3*3*3)
%e A347437                                                 (2*2*2*2*2*2)
%t A347437 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347437 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347437 Table[Length[Select[facs[n],IntegerQ@*altprod]],{n,100}]
%o A347437 (PARI) A347437(n, m=n, ap=1, e=0) = if(1==n, if(e%2, 1==denominator(ap), 1==numerator(ap)), sumdiv(n, d, if((d>1)&&(d<=m), A347437(n/d, d, ap * d^((-1)^e), 1-e)))); \\ _Antti Karttunen_, Oct 22 2023
%Y A347437 Positions of 1's are A005117, complement A013929.
%Y A347437 Allowing any alternating product <= 1 gives A339846.
%Y A347437 Allowing any alternating product > 1 gives A339890.
%Y A347437 The restriction to powers of 2 is A344607.
%Y A347437 The even-length case is A347438, also the case of alternating product 1.
%Y A347437 The reciprocal version is A347439.
%Y A347437 Allowing any alternating product < 1 gives A347440.
%Y A347437 The odd-length case is A347441.
%Y A347437 The reverse version is A347442.
%Y A347437 The additive version is A347446, ranked by A347457.
%Y A347437 Allowing any alternating product >= 1 gives A347456.
%Y A347437 The restriction to perfect squares is A347458, reciprocal A347459.
%Y A347437 The ordered version is A347463.
%Y A347437 A001055 counts factorizations.
%Y A347437 A046099 counts factorizations with no alternating permutations.
%Y A347437 A071321 gives the alternating sum of prime factors of n (reverse: A071322).
%Y A347437 A273013 counts ordered factorizations of n^2 with alternating product 1.
%Y A347437 A347460 counts possible alternating products of factorizations.
%Y A347437 Cf. A025047, A038548, A062312, A088218, A119620, A316523, A330972, A332269, A347445, A347447, A347451, A347454.
%K A347437 nonn
%O A347437 1,4
%A A347437 _Gus Wiseman_, Sep 06 2021
%E A347437 Data section extended up to a(108) by _Antti Karttunen_, Oct 22 2023