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%I #6 Oct 27 2021 22:22:39
%S 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,2,2,1,2,1,4,1,1,
%T 1,6,1,1,1,3,1,2,1,2,2,1,1,6,2,2,1,2,1,3,1,3,1,1,1,5,1,1,2,8,1,2,1,2,
%U 1,2,1,8,1,1,2,2,1,2,1,6,4,1,1,5,1,1,1
%N Number of factorizations of n with alternating product >= 1.
%C We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C Also the number of factorizations of n with alternating sum >= 0.
%F a(n) = A347438(n) + A347440(n).
%e The a(n) factorizations for n = 4, 16, 24, 36, 60, 64, 96:
%e 4 16 24 36 60 64 96
%e 2*2 4*4 2*2*6 6*6 2*5*6 8*8 2*6*8
%e 2*2*4 2*3*4 2*2*9 3*4*5 2*4*8 3*4*8
%e 2*2*2*2 2*3*6 2*2*15 4*4*4 4*4*6
%e 3*3*4 2*3*10 2*2*16 2*2*24
%e 2*2*3*3 2*2*4*4 2*3*16
%e 2*2*2*2*4 2*4*12
%e 2*2*2*2*2*2 2*2*2*2*6
%e 2*2*2*3*4
%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],altprod[#]>=1&]],{n,100}]
%Y The case of partitions is A000041, reverse A344607.
%Y The reverse version is A001055, strict A347705.
%Y Positions of 3's appear to be A065036.
%Y Positions of 1's are 1 and A167171.
%Y The opposite version (<= instead of >=) is A339846.
%Y The strict version (> instead of >=) is A339890, also the odd-length case.
%Y Allowing any integer alternating product gives A347437.
%Y The case of alternating product 1 is A347438, also the even-length case.
%Y Allowing any integer reciprocal alternating product gives A347439.
%Y The complement (< instead of >=) is A347440.
%Y Allowing any integer reverse-alternating product gives A347442.
%Y A038548 counts factorizations with a wiggly permutation.
%Y A045778 counts strict factorizations.
%Y A074206 counts ordered factorizations.
%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A119620 counts partitions with alternating product 1.
%Y A347447 counts strict factorizations with alternating product > 1.
%Y Cf. A001700, A028983, A316523, A347441, A347443, A347446, A347448, A347450, A347454, A347463, A347708.
%K nonn
%O 1,4
%A _Gus Wiseman_, Oct 09 2021
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