A347463 - OEIS

%I #14 Jul 28 2024 10:05:43

%S 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,

%T 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,

%U 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

%N Number of ordered factorizations of n with integer alternating product.

%C An ordered factorization of n is a sequence of positive integers > 1 with product n.

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

%H Antti Karttunen, <a href="/A347463/b347463.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = A347048(n) + A347049(n).

%e The ordered factorizations for n = 4, 8, 12, 16, 24, 32, 36:

%e   4     8       12      16        24      32          36

%e   2*2   4*2     6*2     4*4       12*2    8*4         6*6

%e         2*2*2   2*2*3   8*2       2*2*6   16*2        12*3

%e                 3*2*2   2*2*4     3*2*4   2*2*8       18*2

%e                         2*4*2     4*2*3   2*4*4       2*2*9

%e                         4*2*2     6*2*2   4*2*4       2*3*6

%e                         2*2*2*2           4*4*2       2*6*3

%e                                           8*2*2       3*2*6

%e                                           2*2*4*2     3*3*4

%e                                           4*2*2*2     3*6*2

%e                                           2*2*2*2*2   4*3*3

%e                                                       6*2*3

%e                                                       6*3*2

%e                                                       9*2*2

%e                                                       2*2*3*3

%e                                                       2*3*3*2

%e                                                       3*2*2*3

%e                                                       3*3*2*2

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

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

%t Table[Length[Select[Join@@Permutations/@facs[n],IntegerQ[altprod[#]]&]],{n,100}]

%o (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

%Y Positions of 2's are A001248.

%Y Positions of 1's are A005117.

%Y The restriction to powers of 2 is A116406.

%Y The even-length case is A347048

%Y The odd-length case is A347049.

%Y The unordered version is A347437, reciprocal A347439, reverse A347442.

%Y The case of partitions is A347446, reverse A347445, ranked by A347457.

%Y A001055 counts factorizations (strict A045778, ordered A074206).

%Y A046099 counts factorizations with no alternating permutations.

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

%Y A119620 counts partitions with alternating product 1, ranked by A028982.

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

%Y A339846 counts even-length factorizations, ordered A174725.

%Y A339890 counts odd-length factorizations, ordered A174726.

%Y A347438 counts factorizations with alternating product 1.

%Y A347460 counts possible alternating products of factorizations.

%Y Cf. A025047, A038548, A138364, A347440, A347441, A347453, A347454, A347456, A347458, A347459, A347464, A347705, A347708.

%K nonn

%O 1,4

%A _Gus Wiseman_, Oct 07 2021

%E Data section extended up to a(100) by _Antti Karttunen_, Jul 28 2024