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 A347441 #12 Oct 22 2023 16:43:19
%S A347441 0,1,1,1,1,1,1,2,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,1,2,2,1,1,1,4,1,1,
%T A347441 1,4,1,1,1,2,1,1,1,2,2,1,1,5,1,2,1,2,1,3,1,2,1,1,1,2,1,1,2,5,1,1,1,2,
%U A347441 1,1,1,6,1,1,2,2,1,1,1,5,2,1,1,2,1,1,1,2,1,3,1,2,1,1,1,6,1,2,2,4,1,1,1,2,1,1,1,7
%N A347441 Number of odd-length factorizations of n with integer alternating product.
%C A347441 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A347441 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%H A347441 Antti Karttunen, <a href="/A347441/b347441.txt">Table of n, a(n) for n = 1..65537</a>
%F A347441 a(2^n) = A027193(n).
%e A347441 The a(n) factorizations for n = 2, 8, 32, 48, 54, 72, 108:
%e A347441 2 8 32 48 54 72 108
%e A347441 2*2*2 2*2*8 2*4*6 2*3*9 2*6*6 2*6*9
%e A347441 2*4*4 3*4*4 3*3*6 3*3*8 3*6*6
%e A347441 2*2*2*2*2 2*2*12 2*2*18 2*2*27
%e A347441 2*2*2*2*3 2*3*12 2*3*18
%e A347441 2*2*2*3*3 3*3*12
%e A347441 2*2*3*3*3
%t A347441 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347441 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347441 Table[Length[Select[facs[n],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]
%o A347441 (PARI) A347441(n, m=n, ap=1, e=0) = if(1==n, (e%2)&&1==denominator(ap), sumdiv(n, d, if((d>1)&&(d<=m), A347441(n/d, d, ap * d^((-1)^e), 1-e)))); \\ _Antti Karttunen_, Oct 22 2023
%Y A347441 The restriction to powers of 2 is A027193.
%Y A347441 Positions of 1's are A167207 = A005117 \/ A001248.
%Y A347441 Allowing any alternating product gives A339890.
%Y A347441 Allowing even-length factorizations gives A347437.
%Y A347441 The even-length instead of odd-length version is A347438.
%Y A347441 The additive version is A347444, ranked by A347453.
%Y A347441 A038548 counts possible reverse-alternating products of factorizations.
%Y A347441 A273013 counts ordered factorizations of n^2 with alternating product 1.
%Y A347441 A339846 counts even-length factorizations.
%Y A347441 A347439 counts factorizations with integer reciprocal alternating product.
%Y A347441 A347440 counts factorizations with alternating product < 1.
%Y A347441 A347442 counts factorizations with integer reverse-alternating product.
%Y A347441 A347456 counts factorizations with alternating product >= 1.
%Y A347441 A347463 counts ordered factorizations with integer alternating product.
%Y A347441 Cf. A062312, A119620, A236913, A330972, A347445, A347446, A347447, A347451, A347458, A347460.
%K A347441 nonn
%O A347441 1,8
%A A347441 _Gus Wiseman_, Sep 07 2021
%E A347441 Data section extended up to a(108) by _Antti Karttunen_, Oct 22 2023