|
|
A347048
|
|
Number of even-length ordered factorizations of n with integer alternating product.
|
|
2
|
|
|
1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 4, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 6, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 11, 0, 0, 0, 1, 0, 0, 0, 11, 0, 0, 1, 1, 0, 0, 0, 6, 3, 0, 0, 1, 0, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,16
|
|
COMMENTS
|
An ordered factorization of n is a 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)).
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
The a(n) ordered factorizations for n = 16, 32, 36, 48, 64, 96:
4*4 8*4 6*6 12*4 8*8 24*4
8*2 16*2 12*3 24*2 16*4 48*2
2*2*2*2 2*2*4*2 18*2 2*2*6*2 32*2 3*2*8*2
4*2*2*2 2*2*3*3 3*2*4*2 2*2*4*4 4*2*6*2
2*3*3*2 4*2*3*2 2*2*8*2 6*2*4*2
3*2*2*3 6*2*2*2 2*4*4*2 8*2*3*2
3*3*2*2 4*2*2*4 12*2*2*2
4*2*4*2 2*2*12*2
4*4*2*2
8*2*2*2
2*2*2*2*2*2
|
|
MATHEMATICA
|
ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[Prepend[#, d]&/@ordfacs[n/d], {d, Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[ordfacs[n], EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]], {n, 100}]
|
|
CROSSREFS
|
Positions of 0's are A005117 \ {2}.
The restriction to powers of 2 is A027306.
Positions of 3's appear to be A030514.
Positions of 1's are 1 and A082293.
Allowing non-integer alternating product gives A174725, unordered A339846.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347050 = factorizations with alternating permutation, complement A347706.
A347437 = factorizations with integer alternating product, reverse A347442.
A347446 = partitions with integer alternating product, reverse A347445.
A347460 counts possible alternating products of factorizations.
Cf. A025047, A038548, A116406, A138364, A347440, A347441, A347454, A347456, A347458, A347459, A347464.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|