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 A347439 #23 Jul 28 2024 10:05:35
%S A347439 1,0,0,1,0,0,0,1,1,0,0,1,0,0,0,3,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,3,0,0,
%T A347439 0,4,0,0,0,1,0,0,0,1,1,0,0,3,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,6,0,0,0,1,
%U A347439 0,0,0,5,0,0,1,1,0,0,0,3,3,0,0,1,0,0,0,1,0,1,0,1,0,0,0,3,0,1,1,4,0,0,0,1,0,0,0,5
%N A347439 Number of factorizations of n with integer reciprocal alternating product.
%C A347439 All of these factorizations have an even number of factors, so their reverse-alternating product is also an integer.
%C A347439 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A347439 We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
%C A347439 The value of a(n) does not depend solely on the prime signature of n. See the example comparing a(144) and a(400). - _Antti Karttunen_, Jul 28 2024
%H A347439 Antti Karttunen, <a href="/A347439/b347439.txt">Table of n, a(n) for n = 1..65537</a>
%F A347439 a(2^n) = A027187(n).
%F A347439 a(n^2) = A347459(n).
%e A347439 The a(n) factorizations for
%e A347439 n = 16, 36, 64, 72, 128, 144:
%e A347439 a(n) = 3, 4, 6, 5, 7, 11
%e A347439 --------------------------------------------------------------------------------
%e A347439 2*8 6*6 8*8 2*36 2*64 2*72
%e A347439 4*4 2*18 2*32 3*24 4*32 3*48
%e A347439 2*2*2*2 3*12 4*16 6*12 8*16 4*36
%e A347439 2*2*3*3 2*2*2*8 2*2*3*6 2*2*4*8 6*24
%e A347439 2*2*4*4 2*3*3*4 2*4*4*4 12*12
%e A347439 2*2*2*2*2*2 2*2*2*16 2*2*6*6
%e A347439 2*2*2*2*2*4 2*3*3*8
%e A347439 3*3*4*4
%e A347439 2*2*2*18
%e A347439 2*2*3*12
%e A347439 2*2*2*2*3*3
%e A347439 From _Antti Karttunen_, Jul 28 2024 (Start)
%e A347439 For n=400, there are 12 such factorizations:
%e A347439 2*200
%e A347439 4*100
%e A347439 5*80
%e A347439 10*40
%e A347439 20*20
%e A347439 2*2*2*50
%e A347439 2*2*5*20
%e A347439 2*2*10*10
%e A347439 2*4*5*10
%e A347439 2*5*5*8
%e A347439 4*4*5*5
%e A347439 2*2*2*2*5*5.
%e A347439 Note that 400 = 2^4 * 5^2 has the same prime signature as 144 = 2^4 * 3^2. 400 = 2*4*5*10 is the factorization for which there is no analogous factorization of 144, as 2*3*4*6 doesn't satisfy the condition of having an integer reciprocal alternating product.
%e A347439 (End)
%t A347439 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347439 recaltprod[q_]:=Product[q[[i]]^(-1)^i,{i,Length[q]}];
%t A347439 Table[Length[Select[facs[n],IntegerQ[recaltprod[#]]&]],{n,100}]
%o A347439 (PARI) A347439(n, m=n, ap=1, e=0) = if(1==n, !(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1 && d<=m, A347439(n/d, d, ap * d^((-1)^e), 1-e)))); \\ _Antti Karttunen_, Jul 28 2024
%o A347439 (PARI) A347439(n, m=0, ap=1, e=1) = if(1==n, 1==denominator(ap), sumdiv(n, d, if(d>1 && d>=m, A347439(n/d, d, ap * d^((-1)^e), 1-e)))); \\ _Antti Karttunen_, Jul 28 2024
%Y A347439 Positions of 0's are A005117 \ {1}.
%Y A347439 Positions of non-0's are 1 and A013929.
%Y A347439 The restriction to powers of 2 is A027187, reverse A035363.
%Y A347439 Positions of 1's are 1 and A082293.
%Y A347439 The additive version is A119620, ranked by A347451 and A028982.
%Y A347439 Allowing any alternating product <= 1 gives A339846.
%Y A347439 Allowing any alternating product > 1 gives A339890.
%Y A347439 The non-reciprocal version is A347437.
%Y A347439 The reverse version is A347438.
%Y A347439 Allowing any alternating product < 1 gives A347440.
%Y A347439 The non-reciprocal reverse version is A347442.
%Y A347439 Allowing any alternating product >= 1 gives A347456.
%Y A347439 The restriction to perfect squares is A347459, non-reciprocal A347458.
%Y A347439 A038548 counts possible reverse-alternating products of factorizations.
%Y A347439 A046099 counts factorizations with no alternating permutations.
%Y A347439 A071321 gives the alternating sum of prime factors (reverse: A071322).
%Y A347439 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A347439 A273013 counts ordered factorizations of n^2 with alternating product 1.
%Y A347439 A347441 counts odd-length factorizations with integer alternating product.
%Y A347439 A347460 counts possible alternating products of factorizations.
%Y A347439 Cf. A236913, A316523, A330972, A332269, A344606, A344607, A347445, A347446, A347454, A347457, A347463.
%K A347439 nonn
%O A347439 1,16
%A A347439 _Gus Wiseman_, Sep 07 2021
%E A347439 Data section extended up to a(108) by _Antti Karttunen_, Jul 28 2024