A347048 - OEIS

%I #18 Jul 28 2024 10:05:23

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

%C An ordered factorization of n is a sequence of positive integers > 1 with product n.

%C We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

%H Antti Karttunen, <a href="/A347048/b347048.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = A347463(n) - A347049(n).

%e The a(n) ordered factorizations for n = 16, 32, 36, 48, 64, 96:

%e   4*4       8*4       6*6       12*4      8*8           24*4

%e   8*2       16*2      12*3      24*2      16*4          48*2

%e   2*2*2*2   2*2*4*2   18*2      2*2*6*2   32*2          3*2*8*2

%e             4*2*2*2   2*2*3*3   3*2*4*2   2*2*4*4       4*2*6*2

%e                       2*3*3*2   4*2*3*2   2*2*8*2       6*2*4*2

%e                       3*2*2*3   6*2*2*2   2*4*4*2       8*2*3*2

%e                       3*3*2*2             4*2*2*4       12*2*2*2

%e                                           4*2*4*2       2*2*12*2

%e                                           4*4*2*2

%e                                           8*2*2*2

%e                                           2*2*2*2*2*2

%t ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];

%t altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];

%t Table[Length[Select[ordfacs[n],EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]

%o (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 Positions of 0's are A005117 \ {2}.

%Y The restriction to powers of 2 is A027306.

%Y Heinz numbers of partitions of this type are A028260 /\ A347457.

%Y Positions of 3's appear to be A030514.

%Y Positions of 1's are 1 and A082293.

%Y Allowing non-integer alternating product gives A174725, unordered A339846.

%Y The odd-length version is A347049.

%Y The unordered version is A347438, reverse A347439.

%Y Allowing any length gives A347463.

%Y Partitions of this type are counted by A347704, reverse A035363.

%Y A001055 counts factorizations (strict A045778, ordered A074206).

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A119620 counts partitions with alternating product 1, ranked by A028982.

%Y A273013 counts ordered factorizations of n^2 with alternating product 1.

%Y A339890 counts odd-length factorizations, ordered A174726.

%Y A347050 = factorizations with alternating permutation, complement A347706.

%Y A347437 = factorizations with integer alternating product, reverse A347442.

%Y A347446 = partitions with integer alternating product, reverse A347445.

%Y A347460 counts possible alternating products of factorizations.

%Y Cf. A025047, A038548, A116406, A138364, A347440, A347441, A347454, A347456, A347458, A347459, A347464.

%K nonn

%O 1,16

%A _Gus Wiseman_, Oct 10 2021

%E Data section extended up to a(105) by _Antti Karttunen_, Jul 28 2024