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

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, 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, 1, 3, 0, 8, 0, 1, 2, 2, 1, 3, 0, 6, 1, 1, 0, 6, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Sep 07 2021

Keywords

• nonn

Comments

All such factorizations have even length and alternating sum < 0, so partitions of this type are counted by A344608.

Also the number of factorizations of n with alternating sum < 0.

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

Examples

			The a(n) factorizations for n = 6, 12, 24, 30, 48, 72, 96, 120:
  2*3  2*6  3*8      5*6   6*8      8*9      2*48         2*60
       3*4  4*6      2*15  2*24     2*36     3*32         3*40
            2*12     3*10  3*16     3*24     4*24         4*30
            2*2*2*3        4*12     4*18     6*16         5*24
                           2*2*2*6  6*12     8*12         6*20
                           2*2*3*4  2*2*2*9  2*2*3*8      8*15
                                    2*2*3*6  2*2*4*6      10*12
                                    2*3*3*4  2*3*4*4      2*2*5*6
                                             2*2*2*12     2*3*4*5
                                             2*2*2*2*2*3  2*2*2*15
                                                          2*2*3*10
		

Links

• PlanetMath, alternating sum

Crossrefs

Positions of 0's are A000430.

Positions of 2's are A054753.

Positions of non-0's are A080257.

Positions of 1's are A332269.

The weak version (<= 1 instead of < 1) is A339846, ranked by A028982.

The reciprocal version is A339890.

The additive version is A344608, ranked by A119899.

The even-sum additive version is A344743, ranked by A119899 /\ A300061.

Allowing any integer alternating product gives A347437, additive A347446.

The equal version (= 1 instead of < 1) is A347438.

Allowing any integer reciprocal alternating product gives A347439.

The complement (>= 1 instead of < 1) is counted by A347456.

A038548 counts possible reverse-alternating products of factorizations.

A046099 counts factorizations with no alternating permutations.

A071321 gives the alternating sum of prime factors (reverse: A071322).

A236913 counts partitions of 2n with reverse-alternating sum <= 0.

A273013 counts ordered factorizations of n^2 with alternating product 1.

A347460 counts possible alternating products of factorizations.

Cf. A008549, A058622, A119620, A294175, A330972, A347442, A347447, A347454.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],altprod[#]<1&]],{n,100}]

Formula

a(2^n) = A344608(n).

a(n) = A339846(n) - A347438(n).