A347440 -id:A347440 - cp's OEIS frontend

A347049 Number of odd-length ordered factorizations of n with integer alternating product.

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, 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, 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

Offset: 1

Gus Wiseman, Oct 10 2021

nonn

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

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

			The a(n) ordered factorizations for n = 2, 8, 12, 16, 24, 32, 36, 48:
  2   8       12      16      24      32          36      48
      2*2*2   2*2*3   2*2*4   2*2*6   2*2*8       2*2*9   2*4*6
              3*2*2   2*4*2   3*2*4   2*4*4       2*3*6   3*2*8
                      4*2*2   4*2*3   4*2*4       2*6*3   3*4*4
                              6*2*2   4*4*2       3*2*6   4*2*6
                                      8*2*2       3*3*4   4*4*3
                                      2*2*2*2*2   3*6*2   6*2*4
                                                  4*3*3   6*4*2
                                                  6*2*3   8*2*3
                                                  6*3*2   12*2*2
                                                  9*2*2   2*2*12
                                                          2*2*2*2*3
                                                          2*2*3*2*2
                                                          3*2*2*2*2

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Positions of 2's appear to be A030078.
Positions of 3's appear to be A054753.
Positions of 1's appear to be A167207.
Allowing non-integer alternating product gives A174726, unordered A339890.
The even-length version is A347048.
The unordered version is A347441, with same reverse version.
The case of partitions is A347444, ranked by A347453.
Allowing any length gives A347463.
A001055 counts factorizations (strict A045778, ordered A074206).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A339846 counts even-length factorizations, ordered A174725.
A347050 = factorizations with alternating permutation, complement A347706.
A347437 = factorizations with integer alternating product, reverse A347442.
A347438 = factorizations with alternating product 1, on squares A273013.
A347439 = factorizations with integer reciprocal alternating product.
A347446 = partitions with integer alternating product, reverse A347445.
A347457 lists Heinz numbers of partitions with integer alternating product.
A347460 counts possible alternating products of factorizations.
A347708 counts possible alternating products of odd-length factorizations.
Cf. A025047, A035363, A038548, A116406, A347440, A347454, A347456, A347458, A347459, A347464, A347704.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[ordfacs[n],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]

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

a(n) = A347463(n) - A347048(n).

Data section extended up to a(100) by Antti Karttunen, Jul 28 2024

A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 0, 4, 0, 0, 1, 1, 0, 1, 0, 3, 1, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 1, 2, 0, 1, 0, 2, 1, 0, 0, 4, 0, 1, 0, 3, 0, 1, 0, 1, 1, 0, 0, 5

Offset: 1

Gus Wiseman, Oct 14 2021

nonn

This is also the number of distinct possible alternating products of length-3 factorizations of n, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)), and where a factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			Representative factorizations for each of the a(360) = 9 alternating products:
   (2,2,90) -> 90
   (2,3,60) -> 40
   (2,4,45) -> 45/2
   (2,5,36) -> 72/5
   (2,6,30) -> 10
   (2,9,20) -> 40/9
  (2,10,18) -> 18/5
  (2,12,15) -> 5/2
   (3,8,15) -> 45/8

Antti Karttunen, Table of n, a(n) for n = 1..65537

Allowing factorizations of any length <= 3 gives A033273.
Positions of positive terms are A033942.
Positions of 0's are A037143.
The length-2 version is A072670.
Allowing any length gives A347460, reverse A038548.
Allowing any odd length gives A347708.
A001055 counts factorizations (strict A045778, ordered A074206).
A122179 counts length-3 factorizations.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions, positive A276024.
Cf. A000040, A001358, A002033, A046951, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456, A347461.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],Length[#]==3&]]],{n,100}]

A347709(n) = { my(rats=List([])); fordiv(n,z,my(yx=n/z); fordiv(yx, y, my(x = yx/y); if((y <= z) && (x <= y) && (x > 1), listput(rats,x*z/y)))); #Set(rats); }; \\ Antti Karttunen, Jan 29 2025

More terms from Antti Karttunen, Jan 29 2025

cp's OEIS Frontend

A347049 Number of odd-length ordered factorizations of n with integer alternating product.

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