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A347459 Number of factorizations of n^2 with integer reciprocal alternating product.

%I A347459 #19 Jul 28 2024 10:05:39
%S A347459 1,1,1,3,1,4,1,6,3,4,1,11,1,4,4,12,1,11,1,12,4,4,1,28,3,4,6,12,1,19,1,
%T A347459 22,4,4,4,38,1,4,4,29,1,21,1,12,11,4,1,65,3,11,4,12,1,29,4,29,4,4,1,
%U A347459 71,1,4,11,40,4,22,1,12,4,18,1,107,1,4,11,12,4,22,1,66,12,4,1,76,4,4,4,30,1,71,4,12,4,4,4,141
%N A347459 Number of factorizations of n^2 with integer reciprocal alternating product.
%C A347459 We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
%C A347459 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A347459 All such factorizations have even length.
%C A347459 Image appears to be 1, 3, 4, 6, 11, ... , missing some numbers such as 2, 5, 7, 8, 9, ...
%C A347459 The case of alternating product 1, the case of alternating sum 0, and the reverse version are all counted by A001055.
%H A347459 Antti Karttunen, <a href="/A347459/b347459.txt">Table of n, a(n) for n = 1..16384</a>
%F A347459 a(2^n) = A236913(n).
%F A347459 a(n) = A347439(n^2).
%e A347459 The a(2) = 1 through a(10) = 4 factorizations:
%e A347459     2*2  3*3  2*8      5*5  6*6      7*7  8*8          9*9      2*50
%e A347459               4*4           2*18          2*32         3*27     5*20
%e A347459               2*2*2*2       3*12          4*16         3*3*3*3  10*10
%e A347459                             2*2*3*3       2*2*2*8               2*2*5*5
%e A347459                                           2*2*4*4
%e A347459                                           2*2*2*2*2*2
%t A347459 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347459 recaltprod[q_]:=Product[q[[i]]^(-1)^i,{i,Length[q]}];
%t A347459 Table[Length[Select[facs[n^2],IntegerQ[recaltprod[#]]&]],{n,100}]
%o A347459 (PARI)
%o A347459 A347439(n, m=n, ap=1, e=0) = if(1==n, !(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1 && d<=m, A347439(n/d, d, ap * d^((-1)^e), 1-e))));
%o A347459 A347459(n) = A347439(n^2); \\ _Antti Karttunen_, Jul 28 2024
%Y A347459 Positions of 1's are 1 and A000040, squares A001248.
%Y A347459 The additive version (partitions) is A000041, the even bisection of A119620.
%Y A347459 Partitions of this type are ranked by A028982 and A347451.
%Y A347459 The restriction to powers of 2 is A236913, the even bisection of A027187.
%Y A347459 The nonsquared nonreciprocal even-length version is A347438.
%Y A347459 This is the restriction to perfect squares of A347439.
%Y A347459 The nonreciprocal version is A347458, non-squared A347437.
%Y A347459 A000290 lists squares, complement A000037.
%Y A347459 A001055 counts factorizations.
%Y A347459 A046099 counts factorizations with no alternating permutations.
%Y A347459 A273013 counts ordered factorizations of n^2 with alternating product 1.
%Y A347459 A347460 counts possible alternating products of factorizations.
%Y A347459 A339846 counts even-length factorizations.
%Y A347459 A339890 counts odd-length factorizations.
%Y A347459 A347457 ranks partitions with integer alternating product.
%Y A347459 A347466 counts factorizations of n^2.
%Y A347459 Cf. A062312, A316523, A330972, A344653, A346635, A347440, A347441, A347442, A347453, A347461, A347463, A347464, A347705.
%K A347459 nonn
%O A347459 1,4
%A A347459 _Gus Wiseman_, Sep 22 2021
%E A347459 Data section extended up to a(96) by _Antti Karttunen_, Jul 28 2024