A347460 Number of distinct possible alternating products of factorizations of n.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 5, 2, 6, 2, 2, 1, 10, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 10, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 10, 2, 2

Offset

1,4

Comments

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

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

Links

Table of n, a(n) for n=1..86.

Example

The a(n) alternating products for n = 1, 4, 8, 12, 24, 30, 36, 48, 60, 120:
  1  4  8    12   24   30    36   48    60    120
     1  2    3    6    10/3  9    12    15    30
        1/2  3/4  8/3  5/6   4    16/3  20/3  40/3
             1/3  2/3  3/10  1    3     15/4  15/2
                  3/8  2/15  4/9  3/4   12/5  24/5
                  1/6        1/4  1/3   3/5   10/3
                             1/9  3/16  5/12  5/6
                                  1/12  4/15  8/15
                                        3/20  3/10
                                        1/15  5/24
                                              2/15
                                              3/40
                                              1/30

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[Union[altprod/@facs[n]]], {n, 100}]

Crossrefs

Positions of 1's are 1 and A000040.

Positions of 2's appear to be A001358.

Positions of 3's appear to be A030078.

Dominates A038548, the version for reverse-alternating product.

Counting only integers gives A046951.

The even-length case is A072670.

The version for partitions (not factorizations) is A347461, reverse A347462.

The odd-length case is A347708.

The length-3 case is A347709.

A001055 counts factorizations (strict A045778, ordered A074206).

A056239 adds up prime indices, row sums of A112798.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A108917 counts knapsack partitions, ranked by A299702.

A276024 counts distinct positive subset-sums of partitions, strict A284640.

A292886 counts knapsack factorizations, by sum A293627.

A299701 counts distinct subset-sums of prime indices, positive A304793.

A301957 counts distinct subset-products of prime indices.

A304792 counts distinct subset-sums of partitions.

Cf. A002033, A119620, A143823, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.

Sequence in context: A076526 A351417 A226378 * A033273 A319685 A343652

Adjacent sequences: A347457 A347458 A347459 * A347461 A347462 A347463

Keyword

nonn

Author

Gus Wiseman, Oct 06 2021

Status

approved