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A347458 Number of factorizations of n^2 with integer alternating product.

%I A347458 #23 Oct 22 2023 16:43:54
%S A347458 1,2,2,4,2,6,2,8,4,6,2,17,2,6,6,15,2,17,2,16,6,6,2,41,4,6,8,16,2,31,2,
%T A347458 27,6,6,6,56,2,6,6,39,2,31,2,17,17,6,2,90,4,17,6,17,2,41,6,39,6,6,2,
%U A347458 105,2,6,17,48,6,31,2,17,6,31,2,148,2,6,17,17,6,32,2,86,15,6,2,107,6,6,6,40,2,109,6,17
%N A347458 Number of factorizations of n^2 with integer alternating product.
%C A347458 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C A347458 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A347458 The even-length case, the case of alternating product 1, and the case of alternating sum 0 are all counted by A001055.
%H A347458 Antti Karttunen, <a href="/A347458/b347458.txt">Table of n, a(n) for n = 1..16415</a>
%F A347458 a(2^n) = A344611(n).
%F A347458 a(n) = A347437(n^2).
%e A347458 The a(2) = 2 through a(8) = 8 factorizations:
%e A347458   4     9     16        25    36        49    64
%e A347458   2*2   3*3   4*4       5*5   6*6       7*7   8*8
%e A347458               2*2*4           2*2*9           2*4*8
%e A347458               2*2*2*2         2*3*6           4*4*4
%e A347458                               3*3*4           2*2*16
%e A347458                               2*2*3*3         2*2*4*4
%e A347458                                               2*2*2*2*4
%e A347458                                               2*2*2*2*2*2
%t A347458 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347458 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347458 Table[Length[Select[facs[n^2],IntegerQ[altprod[#]]&]],{n,100}]
%o A347458 (PARI)
%o A347458 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))));
%o A347458 A347458(n) = A347437(n*n); \\ _Antti Karttunen_, Oct 22 2023
%Y A347458 Positions of 2's are A000040, squares A001248.
%Y A347458 The restriction to powers of 2 is A344611.
%Y A347458 This is the restriction to perfect squares of A347437.
%Y A347458 The nonsquared even-length version is A347438.
%Y A347458 The reciprocal version is A347459, non-squared A347439.
%Y A347458 The additive version (partitions) is the even bisection of A347446.
%Y A347458 The nonsquared ordered version is A347463.
%Y A347458 The case of alternating product 1 in the ordered version is A347464.
%Y A347458 Allowing any alternating product gives A347466.
%Y A347458 A000290 lists squares, complement A000037.
%Y A347458 A001055 counts factorizations.
%Y A347458 A046099 counts factorizations with no alternating permutations.
%Y A347458 A071321 gives the alternating sum of prime factors of n (reverse: A071322).
%Y A347458 A273013 counts ordered factorizations of n^2 with alternating product 1.
%Y A347458 A347460 counts possible alternating products of factorizations.
%Y A347458 A339846 counts even-length factorizations.
%Y A347458 A339890 counts odd-length factorizations.
%Y A347458 A347457 ranks partitions with integer alternating product.
%Y A347458 Cf. A062312, A119620, A330972, A346635, A347440, A347441, A347442, A347445, A347451, A347456, A347704, A347705.
%Y A347458 Apparently, A006881 gives the positions of 6's. - _Antti Karttunen_, Oct 22 2023
%K A347458 nonn
%O A347458 1,2
%A A347458 _Gus Wiseman_, Sep 21 2021
%E A347458 Data section extended up to a(92) by _Antti Karttunen_, Oct 22 2023