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 A347049 #15 Jul 28 2024 10:05:31
%S A347049 0,1,1,1,1,1,1,2,1,1,1,3,1,1,1,4,1,3,1,3,1,1,1,5,1,1,2,3,1,1,1,7,1,1,
%T A347049 1,11,1,1,1,5,1,1,1,3,3,1,1,14,1,3,1,3,1,5,1,5,1,1,1,7,1,1,3,15,1,1,1,
%U A347049 3,1,1,1,24,1,1,3,3,1,1,1,14,4,1,1,7,1,1,1,5,1,7,1,3,1,1,1,24,1,3,3,11
%N A347049 Number of odd-length ordered factorizations of n with integer alternating product.
%C A347049 An ordered factorization of n is a sequence of positive integers > 1 with product n.
%C A347049 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%H A347049 Antti Karttunen, <a href="/A347049/b347049.txt">Table of n, a(n) for n = 1..65537</a>
%H A347049 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A347049 a(n) = A347463(n) - A347048(n).
%e A347049 The a(n) ordered factorizations for n = 2, 8, 12, 16, 24, 32, 36, 48:
%e A347049 2 8 12 16 24 32 36 48
%e A347049 2*2*2 2*2*3 2*2*4 2*2*6 2*2*8 2*2*9 2*4*6
%e A347049 3*2*2 2*4*2 3*2*4 2*4*4 2*3*6 3*2*8
%e A347049 4*2*2 4*2*3 4*2*4 2*6*3 3*4*4
%e A347049 6*2*2 4*4*2 3*2*6 4*2*6
%e A347049 8*2*2 3*3*4 4*4*3
%e A347049 2*2*2*2*2 3*6*2 6*2*4
%e A347049 4*3*3 6*4*2
%e A347049 6*2*3 8*2*3
%e A347049 6*3*2 12*2*2
%e A347049 9*2*2 2*2*12
%e A347049 2*2*2*2*3
%e A347049 2*2*3*2*2
%e A347049 3*2*2*2*2
%t A347049 ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
%t A347049 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347049 Table[Length[Select[ordfacs[n],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]
%o A347049 (PARI) A347049(n, m=n, ap=1, e=0) = if(1==n,(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1, A347049(n/d, d, ap * d^((-1)^e), 1-e)))); \\ _Antti Karttunen_, Jul 28 2024
%Y A347049 Positions of 2's appear to be A030078.
%Y A347049 Positions of 3's appear to be A054753.
%Y A347049 Positions of 1's appear to be A167207.
%Y A347049 Allowing non-integer alternating product gives A174726, unordered A339890.
%Y A347049 The even-length version is A347048.
%Y A347049 The unordered version is A347441, with same reverse version.
%Y A347049 The case of partitions is A347444, ranked by A347453.
%Y A347049 Allowing any length gives A347463.
%Y A347049 A001055 counts factorizations (strict A045778, ordered A074206).
%Y A347049 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A347049 A119620 counts partitions with alternating product 1, ranked by A028982.
%Y A347049 A339846 counts even-length factorizations, ordered A174725.
%Y A347049 A347050 = factorizations with alternating permutation, complement A347706.
%Y A347049 A347437 = factorizations with integer alternating product, reverse A347442.
%Y A347049 A347438 = factorizations with alternating product 1, on squares A273013.
%Y A347049 A347439 = factorizations with integer reciprocal alternating product.
%Y A347049 A347446 = partitions with integer alternating product, reverse A347445.
%Y A347049 A347457 lists Heinz numbers of partitions with integer alternating product.
%Y A347049 A347460 counts possible alternating products of factorizations.
%Y A347049 A347708 counts possible alternating products of odd-length factorizations.
%Y A347049 Cf. A025047, A035363, A038548, A116406, A347440, A347454, A347456, A347458, A347459, A347464, A347704.
%K A347049 nonn
%O A347049 1,8
%A A347049 _Gus Wiseman_, Oct 10 2021
%E A347049 Data section extended up to a(100) by _Antti Karttunen_, Jul 28 2024