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%I A347447 #8 Sep 27 2021 07:55:36
%S A347447 0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,
%T A347447 1,2,1,1,1,2,1,2,1,1,1,1,1,3,1,1,1,1,1,2,1,2,1,1,1,4,1,1,1,2,1,2,1,1,
%U A347447 1,2,1,4,1,1,1,1,1,2,1,3,1,1,1,4,1,1,1
%N A347447 Number of strict factorizations of n with alternating product > 1.
%C A347447 A strict factorization of n is an increasing sequence of distinct positive integers > 1 with product n.
%C A347447 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C A347447 All such factorizations must have odd length.
%e A347447 The a(720) = 30 factorizations:
%e A347447   (2*4*90)     (3*4*60)   (4*5*36)   (5*6*24)  (6*8*15)   (8*9*10)  (720)
%e A347447   (2*5*72)     (3*5*48)   (4*6*30)   (5*8*18)  (6*10*12)
%e A347447   (2*6*60)     (3*6*40)   (4*9*20)   (5*9*16)
%e A347447   (2*8*45)     (3*8*30)   (4*10*18)
%e A347447   (2*9*40)     (3*10*24)  (4*12*15)
%e A347447   (2*10*36)    (3*12*20)
%e A347447   (2*12*30)    (3*15*16)
%e A347447   (2*15*24)
%e A347447   (2*18*20)
%e A347447   (2*3*120)
%e A347447   (2*3*4*5*6)
%t A347447 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347447 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347447 Table[Length[Select[facs[n],UnsameQ@@#&&altprod[#]>1&]],{n,100}]
%Y A347447 Allowing any alternating product gives A045778.
%Y A347447 The reverse additive version (or restriction to powers of 2) is A067659.
%Y A347447 The non-strict version is A339890.
%Y A347447 Allowing equal parts and any alternating product < 1 gives A347440.
%Y A347447 Allowing equal parts and any alternating product >= 1 gives A347456.
%Y A347447 A046099 counts factorizations with no alternating permutations.
%Y A347447 A273013 counts ordered factorizations of n^2 with alternating product 1.
%Y A347447 A339846 counts even-length factorizations.
%Y A347447 A347437 counts factorizations with integer alternating product.
%Y A347447 A347441 counts odd-length factorizations with integer alternating product.
%Y A347447 A347460 counts possible alternating products of factorizations.
%Y A347447 Cf. A000009, A005117, A030059, A119620, A119899, A330972, A344608, A347438, A347439, A347442, A347463.
%K A347447 nonn
%O A347447 1,24
%A A347447 _Gus Wiseman_, Sep 23 2021

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