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A347463
Number of ordered factorizations of n with integer alternating product.
26
1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 7, 1, 4, 1, 4, 1, 1, 1, 6, 2, 1, 3, 4, 1, 1, 1, 11, 1, 1, 1, 18, 1, 1, 1, 6, 1, 1, 1, 4, 4, 1, 1, 20, 2, 4, 1, 4, 1, 6, 1, 6, 1, 1, 1, 8, 1, 1, 4, 26, 1, 1, 1, 4, 1, 1, 1, 35, 1, 1, 4, 4, 1, 1, 1, 20, 7, 1, 1, 8, 1, 1, 1, 6, 1, 8, 1, 4, 1, 1, 1, 32, 1, 4, 4, 18
OFFSET
1,4
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)).
FORMULA
a(n) = A347048(n) + A347049(n).
EXAMPLE
The ordered factorizations for n = 4, 8, 12, 16, 24, 32, 36:
4 8 12 16 24 32 36
2*2 4*2 6*2 4*4 12*2 8*4 6*6
2*2*2 2*2*3 8*2 2*2*6 16*2 12*3
3*2*2 2*2*4 3*2*4 2*2*8 18*2
2*4*2 4*2*3 2*4*4 2*2*9
4*2*2 6*2*2 4*2*4 2*3*6
2*2*2*2 4*4*2 2*6*3
8*2*2 3*2*6
2*2*4*2 3*3*4
4*2*2*2 3*6*2
2*2*2*2*2 4*3*3
6*2*3
6*3*2
9*2*2
2*2*3*3
2*3*3*2
3*2*2*3
3*3*2*2
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[Join@@Permutations/@facs[n], IntegerQ[altprod[#]]&]], {n, 100}]
PROG
(PARI) A347463(n, m=n, ap=1, e=0) = if(1==n, if(e%2, 1==denominator(ap), 1==numerator(ap)), sumdiv(n, d, if(d>1, A347463(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024
CROSSREFS
Positions of 2's are A001248.
Positions of 1's are A005117.
The restriction to powers of 2 is A116406.
The even-length case is A347048
The odd-length case is A347049.
The unordered version is A347437, reciprocal A347439, reverse A347442.
The case of partitions is A347446, reverse A347445, ranked by A347457.
A001055 counts factorizations (strict A045778, ordered A074206).
A046099 counts factorizations with no alternating permutations.
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.
A339846 counts even-length factorizations, ordered A174725.
A339890 counts odd-length factorizations, ordered A174726.
A347438 counts factorizations with alternating product 1.
A347460 counts possible alternating products of factorizations.
Sequence in context: A316978 A331023 A284345 * A373378 A337619 A183214
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 07 2021
EXTENSIONS
Data section extended up to a(100) by Antti Karttunen, Jul 28 2024
STATUS
approved