A347439 Number of factorizations of n with integer reciprocal alternating product.

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, 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, 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

Offset

1,16

Comments

All of these factorizations have an even number of factors, so their reverse-alternating product is also an integer.

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

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

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

Links

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

Formula

a(2^n) = A027187(n).

a(n^2) = A347459(n).

Example

The a(n) factorizations for
n    = 16,       36,       64,           72,       128,          144:
a(n) = 3,        4,        6,            5,        7,            11
--------------------------------------------------------------------------------
       2*8       6*6       8*8           2*36      2*64          2*72
       4*4       2*18      2*32          3*24      4*32          3*48
       2*2*2*2   3*12      4*16          6*12      8*16          4*36
                 2*2*3*3   2*2*2*8       2*2*3*6   2*2*4*8       6*24
                           2*2*4*4       2*3*3*4   2*4*4*4       12*12
                           2*2*2*2*2*2             2*2*2*16      2*2*6*6
                                                   2*2*2*2*2*4   2*3*3*8
                                                                 3*3*4*4
                                                                 2*2*2*18
                                                                 2*2*3*12
                                                                 2*2*2*2*3*3
From Antti Karttunen, Jul 28 2024 (Start)
For n=400, there are 12 such factorizations:
  2*200
  4*100
  5*80
  10*40
  20*20
  2*2*2*50
  2*2*5*20
  2*2*10*10
  2*4*5*10
  2*5*5*8
  4*4*5*5
  2*2*2*2*5*5.
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.
(End)

Mathematica

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

Prog

(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
(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

Crossrefs

Positions of 0's are A005117 \ {1}.

Positions of non-0's are 1 and A013929.

The restriction to powers of 2 is A027187, reverse A035363.

Positions of 1's are 1 and A082293.

The additive version is A119620, ranked by A347451 and A028982.

Allowing any alternating product <= 1 gives A339846.

Allowing any alternating product > 1 gives A339890.

The non-reciprocal version is A347437.

The reverse version is A347438.

Allowing any alternating product < 1 gives A347440.

The non-reciprocal reverse version is A347442.

Allowing any alternating product >= 1 gives A347456.

The restriction to perfect squares is A347459, non-reciprocal A347458.

A038548 counts possible reverse-alternating products of factorizations.

A046099 counts factorizations with no alternating permutations.

A071321 gives the alternating sum of prime factors (reverse: A071322).

A316524 gives the alternating sum of prime indices (reverse: A344616).

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

A347441 counts odd-length factorizations with integer alternating product.

A347460 counts possible alternating products of factorizations.

Cf. A236913, A316523, A330972, A332269, A344606, A344607, A347445, A347446, A347454, A347457, A347463.

Sequence in context: A318499 A346012 A317946 * A347048 A374213 A140807

Adjacent sequences: A347436 A347437 A347438 * A347440 A347441 A347442

Keyword

nonn

Author

Gus Wiseman, Sep 07 2021

Extensions

Data section extended up to a(108) by Antti Karttunen, Jul 28 2024

Status

approved