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A347048 Number of even-length ordered factorizations of n with integer alternating product.

%I A347048 #18 Jul 28 2024 10:05:23
%S A347048 1,0,0,1,0,0,0,1,1,0,0,1,0,0,0,3,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,4,0,0,
%T A347048 0,7,0,0,0,1,0,0,0,1,1,0,0,6,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,11,0,0,0,1,
%U A347048 0,0,0,11,0,0,1,1,0,0,0,6,3,0,0,1,0,0,0,1,0,1,0,1,0,0,0,8,0,1,1,7,0,0,0,1,0
%N A347048 Number of even-length ordered factorizations of n with integer alternating product.
%C A347048 An ordered factorization of n is a sequence of positive integers > 1 with product n.
%C A347048 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%H A347048 Antti Karttunen, <a href="/A347048/b347048.txt">Table of n, a(n) for n = 1..65537</a>
%H A347048 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A347048 a(n) = A347463(n) - A347049(n).
%e A347048 The a(n) ordered factorizations for n = 16, 32, 36, 48, 64, 96:
%e A347048   4*4       8*4       6*6       12*4      8*8           24*4
%e A347048   8*2       16*2      12*3      24*2      16*4          48*2
%e A347048   2*2*2*2   2*2*4*2   18*2      2*2*6*2   32*2          3*2*8*2
%e A347048             4*2*2*2   2*2*3*3   3*2*4*2   2*2*4*4       4*2*6*2
%e A347048                       2*3*3*2   4*2*3*2   2*2*8*2       6*2*4*2
%e A347048                       3*2*2*3   6*2*2*2   2*4*4*2       8*2*3*2
%e A347048                       3*3*2*2             4*2*2*4       12*2*2*2
%e A347048                                           4*2*4*2       2*2*12*2
%e A347048                                           4*4*2*2
%e A347048                                           8*2*2*2
%e A347048                                           2*2*2*2*2*2
%t A347048 ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
%t A347048 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347048 Table[Length[Select[ordfacs[n],EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]
%o A347048 (PARI) A347048(n, m=n, ap=1, e=0) = if(1==n,!(e%2) && 1==numerator(ap), sumdiv(n, d, if(d>1, A347048(n/d, d, ap * d^((-1)^e), 1-e)))); \\ _Antti Karttunen_, Jul 28 2024
%Y A347048 Positions of 0's are A005117 \ {2}.
%Y A347048 The restriction to powers of 2 is A027306.
%Y A347048 Heinz numbers of partitions of this type are A028260 /\ A347457.
%Y A347048 Positions of 3's appear to be A030514.
%Y A347048 Positions of 1's are 1 and A082293.
%Y A347048 Allowing non-integer alternating product gives A174725, unordered A339846.
%Y A347048 The odd-length version is A347049.
%Y A347048 The unordered version is A347438, reverse A347439.
%Y A347048 Allowing any length gives A347463.
%Y A347048 Partitions of this type are counted by A347704, reverse A035363.
%Y A347048 A001055 counts factorizations (strict A045778, ordered A074206).
%Y A347048 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A347048 A119620 counts partitions with alternating product 1, ranked by A028982.
%Y A347048 A273013 counts ordered factorizations of n^2 with alternating product 1.
%Y A347048 A339890 counts odd-length factorizations, ordered A174726.
%Y A347048 A347050 = factorizations with alternating permutation, complement A347706.
%Y A347048 A347437 = factorizations with integer alternating product, reverse A347442.
%Y A347048 A347446 = partitions with integer alternating product, reverse A347445.
%Y A347048 A347460 counts possible alternating products of factorizations.
%Y A347048 Cf. A025047, A038548, A116406, A138364, A347440, A347441, A347454, A347456, A347458, A347459, A347464.
%K A347048 nonn
%O A347048 1,16
%A A347048 _Gus Wiseman_, Oct 10 2021
%E A347048 Data section extended up to a(105) by _Antti Karttunen_, Jul 28 2024