This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A347463 #14 Jul 28 2024 10:05:43
%S A347463 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 A347463 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 A347463 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 A347463 Number of ordered factorizations of n with integer alternating product.
%C A347463 An ordered factorization of n is a sequence of positive integers > 1 with product n.
%C A347463 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%H A347463 Antti Karttunen, <a href="/A347463/b347463.txt">Table of n, a(n) for n = 1..16384</a>
%H A347463 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A347463 a(n) = A347048(n) + A347049(n).
%e A347463 The ordered factorizations for n = 4, 8, 12, 16, 24, 32, 36:
%e A347463 4 8 12 16 24 32 36
%e A347463 2*2 4*2 6*2 4*4 12*2 8*4 6*6
%e A347463 2*2*2 2*2*3 8*2 2*2*6 16*2 12*3
%e A347463 3*2*2 2*2*4 3*2*4 2*2*8 18*2
%e A347463 2*4*2 4*2*3 2*4*4 2*2*9
%e A347463 4*2*2 6*2*2 4*2*4 2*3*6
%e A347463 2*2*2*2 4*4*2 2*6*3
%e A347463 8*2*2 3*2*6
%e A347463 2*2*4*2 3*3*4
%e A347463 4*2*2*2 3*6*2
%e A347463 2*2*2*2*2 4*3*3
%e A347463 6*2*3
%e A347463 6*3*2
%e A347463 9*2*2
%e A347463 2*2*3*3
%e A347463 2*3*3*2
%e A347463 3*2*2*3
%e A347463 3*3*2*2
%t A347463 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347463 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347463 Table[Length[Select[Join@@Permutations/@facs[n],IntegerQ[altprod[#]]&]],{n,100}]
%o A347463 (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 A347463 Positions of 2's are A001248.
%Y A347463 Positions of 1's are A005117.
%Y A347463 The restriction to powers of 2 is A116406.
%Y A347463 The even-length case is A347048
%Y A347463 The odd-length case is A347049.
%Y A347463 The unordered version is A347437, reciprocal A347439, reverse A347442.
%Y A347463 The case of partitions is A347446, reverse A347445, ranked by A347457.
%Y A347463 A001055 counts factorizations (strict A045778, ordered A074206).
%Y A347463 A046099 counts factorizations with no alternating permutations.
%Y A347463 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A347463 A119620 counts partitions with alternating product 1, ranked by A028982.
%Y A347463 A273013 counts ordered factorizations of n^2 with alternating product 1.
%Y A347463 A339846 counts even-length factorizations, ordered A174725.
%Y A347463 A339890 counts odd-length factorizations, ordered A174726.
%Y A347463 A347438 counts factorizations with alternating product 1.
%Y A347463 A347460 counts possible alternating products of factorizations.
%Y A347463 Cf. A025047, A038548, A138364, A347440, A347441, A347453, A347454, A347456, A347458, A347459, A347464, A347705, A347708.
%K A347463 nonn
%O A347463 1,4
%A A347463 _Gus Wiseman_, Oct 07 2021
%E A347463 Data section extended up to a(100) by _Antti Karttunen_, Jul 28 2024