%I #16 Oct 22 2023 16:43:04

%S 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 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 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 Number of factorizations of n with integer alternating product.

%C A 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="/A347437/b347437.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) = A344607(n).

%F a(n^2) = A347458(n).

%e The factorizations for n = 4, 16, 36, 48, 54, 64, 108:

%e (4) (16) (36) (48) (54) (64) (108)

%e (2*2) (4*4) (6*6) (2*4*6) (2*3*9) (8*8) (2*6*9)

%e (2*2*4) (2*2*9) (3*4*4) (3*3*6) (2*4*8) (3*6*6)

%e (2*2*2*2) (2*3*6) (2*2*12) (4*4*4) (2*2*27)

%e (3*3*4) (2*2*2*2*3) (2*2*16) (2*3*18)

%e (2*2*3*3) (2*2*4*4) (3*3*12)

%e (2*2*2*2*4) (2*2*3*3*3)

%e (2*2*2*2*2*2)

%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],IntegerQ@*altprod]],{n,100}]

%o (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 Positions of 1's are A005117, complement A013929.

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

%Y Allowing any alternating product > 1 gives A339890.

%Y The restriction to powers of 2 is A344607.

%Y The even-length case is A347438, also the case of alternating product 1.

%Y The reciprocal version is A347439.

%Y Allowing any alternating product < 1 gives A347440.

%Y The odd-length case is A347441.

%Y The reverse version is A347442.

%Y The additive version is A347446, ranked by A347457.

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

%Y The restriction to perfect squares is A347458, reciprocal A347459.

%Y The ordered version is A347463.

%Y A001055 counts factorizations.

%Y A046099 counts factorizations with no alternating permutations.

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

%Y A273013 counts ordered factorizations of n^2 with alternating product 1.

%Y A347460 counts possible alternating products of factorizations.

%Y Cf. A025047, A038548, A062312, A088218, A119620, A316523, A330972, A332269, A347445, A347447, A347451, A347454.

%K nonn

%O 1,4

%A _Gus Wiseman_, Sep 06 2021

%E Data section extended up to a(108) by _Antti Karttunen_, Oct 22 2023