A347705 - OEIS

A347705

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

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

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OFFSET

1,6

COMMENTS

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

LINKS

Table of n, a(n) for n=1..85.

FORMULA

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

EXAMPLE

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

2 6 8 12 24 30 48 60

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

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

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

2*2*6 2*3*5 4*12 5*12

2*3*4 2*3*8 6*10

2*2*2*3 2*4*6 2*5*6

3*4*4 3*4*5

2*2*12 2*2*15

2*2*2*6 2*3*10

2*2*3*4 2*2*3*5

2*2*2*2*3

MATHEMATICA

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

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

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

CROSSREFS

Positions of 1's are A000430.

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

The non-reverse version is A339890, strict A347447.

The version for reverse-alternating product 1 is A347438.

Allowing any integer reciprocal alternating product gives A347439.

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

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

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

A001055 counts factorizations (strict A045778, ordered A074206).

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

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

A292886 counts knapsack factorizations, by sum A293627.

A347707 counts possible integer reverse-alternating products of partitions.

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

Sequence in context: A164677 A001511 A265331 * A244569 A266928 A285324

Adjacent sequences: A347702 A347703 A347704 * A347706 A347707 A347708

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 12 2021

STATUS

approved