cp's OEIS Frontend

A347440 Number of factorizations of n with alternating product < 1.

%I A347440 #10 Sep 15 2021 09:52:21
%S A347440 0,0,0,0,0,1,0,1,0,1,0,2,0,1,1,1,0,2,0,2,1,1,0,4,0,1,1,2,0,3,0,3,1,1,
%T A347440 1,3,0,1,1,4,0,3,0,2,2,1,0,6,0,2,1,2,0,4,1,4,1,1,0,6,0,1,2,3,1,3,0,2,
%U A347440 1,3,0,8,0,1,2,2,1,3,0,6,1,1,0,6,1,1,1
%N A347440 Number of factorizations of n with alternating product < 1.
%C A347440 All such factorizations have even length and alternating sum < 0, so partitions of this type are counted by A344608.
%C A347440 Also the number of factorizations of n with alternating sum < 0.
%C A347440 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A347440 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%H A347440 PlanetMath, <a href="https://planetmath.org/alternatingsum">alternating sum</a>
%F A347440 a(2^n) = A344608(n).
%F A347440 a(n) = A339846(n) - A347438(n).
%e A347440 The a(n) factorizations for n = 6, 12, 24, 30, 48, 72, 96, 120:
%e A347440   2*3  2*6  3*8      5*6   6*8      8*9      2*48         2*60
%e A347440        3*4  4*6      2*15  2*24     2*36     3*32         3*40
%e A347440             2*12     3*10  3*16     3*24     4*24         4*30
%e A347440             2*2*2*3        4*12     4*18     6*16         5*24
%e A347440                            2*2*2*6  6*12     8*12         6*20
%e A347440                            2*2*3*4  2*2*2*9  2*2*3*8      8*15
%e A347440                                     2*2*3*6  2*2*4*6      10*12
%e A347440                                     2*3*3*4  2*3*4*4      2*2*5*6
%e A347440                                              2*2*2*12     2*3*4*5
%e A347440                                              2*2*2*2*2*3  2*2*2*15
%e A347440                                                           2*2*3*10
%t A347440 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347440 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347440 Table[Length[Select[facs[n],altprod[#]<1&]],{n,100}]
%Y A347440 Positions of 0's are A000430.
%Y A347440 Positions of 2's are A054753.
%Y A347440 Positions of non-0's are A080257.
%Y A347440 Positions of 1's are A332269.
%Y A347440 The weak version (<= 1 instead of < 1) is A339846, ranked by A028982.
%Y A347440 The reciprocal version is A339890.
%Y A347440 The additive version is A344608, ranked by A119899.
%Y A347440 The even-sum additive version is A344743, ranked by A119899 /\ A300061.
%Y A347440 Allowing any integer alternating product gives A347437, additive A347446.
%Y A347440 The equal version (= 1 instead of < 1) is A347438.
%Y A347440 Allowing any integer reciprocal alternating product gives A347439.
%Y A347440 The complement (>= 1 instead of < 1) is counted by A347456.
%Y A347440 A038548 counts possible reverse-alternating products of factorizations.
%Y A347440 A046099 counts factorizations with no alternating permutations.
%Y A347440 A071321 gives the alternating sum of prime factors (reverse: A071322).
%Y A347440 A236913 counts partitions of 2n with reverse-alternating sum <= 0.
%Y A347440 A273013 counts ordered factorizations of n^2 with alternating product 1.
%Y A347440 A347460 counts possible alternating products of factorizations.
%Y A347440 Cf. A008549, A058622, A119620, A294175, A330972, A347442, A347447, A347454.
%K A347440 nonn
%O A347440 1,12
%A A347440 _Gus Wiseman_, Sep 07 2021