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Number of factorizations of n with reverse-alternating product > 1.
10

%I #6 Oct 27 2021 22:23:57

%S 0,1,1,1,1,2,1,3,1,2,1,4,1,2,2,3,1,4,1,4,2,2,1,7,1,2,3,4,1,5,1,7,2,2,

%T 2,7,1,2,2,7,1,5,1,4,4,2,1,12,1,4,2,4,1,7,2,7,2,2,1,11,1,2,4,8,2,5,1,

%U 4,2,5,1,16,1,2,4,4,2,5,1,12,3,2,1,11,2

%N Number of factorizations of n with reverse-alternating product > 1.

%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)). The reverse-alternating product is the alternating product of the reversed sequence.

%F a(n) = A001055(n) - A347438(n).

%e The a(n) factorizations for n = 2, 6, 8, 12, 24, 30, 48, 60:

%e 2 6 8 12 24 30 48 60

%e 2*3 2*4 2*6 3*8 5*6 6*8 2*30

%e 2*2*2 3*4 4*6 2*15 2*24 3*20

%e 2*2*3 2*12 3*10 3*16 4*15

%e 2*2*6 2*3*5 4*12 5*12

%e 2*3*4 2*3*8 6*10

%e 2*2*2*3 2*4*6 2*5*6

%e 3*4*4 3*4*5

%e 2*2*12 2*2*15

%e 2*2*2*6 2*3*10

%e 2*2*3*4 2*2*3*5

%e 2*2*2*2*3

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t revaltprod[q_]:=Product[q[[-i]]^(-1)^(i-1),{i,Length[q]}];

%t Table[Length[Select[facs[n],revaltprod[#]>1&]],{n,100}]

%Y Positions of 1's are A000430.

%Y The weak version (>= instead of >) is A001055, non-reverse A347456.

%Y The non-reverse version is A339890, strict A347447.

%Y The version for reverse-alternating product 1 is A347438.

%Y Allowing any integer reciprocal alternating product gives A347439.

%Y The even-length case is A347440, also the opposite reverse version.

%Y Allowing any integer rev-alt product gives A347442, non-reverse A347437.

%Y The version for partitions is A347449, non-reverse A347448.

%Y A001055 counts factorizations (strict A045778, ordered A074206).

%Y A038548 counts possible rev-alt products of factorizations, integer A046951.

%Y A103919 counts partitions by sum and alternating sum, reverse A344612.

%Y A292886 counts knapsack factorizations, by sum A293627.

%Y A347707 counts possible integer reverse-alternating products of partitions.

%Y Cf. A028983, A119620, A339846, A347441, A347443, A347450, A347463, A347466.

%K nonn

%O 1,6

%A _Gus Wiseman_, Oct 12 2021