A347437 Number of factorizations of n with integer alternating product.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 4, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 6, 1, 2, 2, 6, 1, 1, 1, 2, 1, 1, 1, 7

Offset

1,4

Comments

A factorization of n is a weakly increasing 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)).

Links

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

PlanetMath, alternating sum

Formula

a(2^n) = A344607(n).

a(n^2) = A347458(n).

Example

The factorizations for n = 4, 16, 36, 48, 54, 64, 108:
  (4)   (16)      (36)      (48)        (54)    (64)          (108)
  (2*2) (4*4)     (6*6)     (2*4*6)     (2*3*9) (8*8)         (2*6*9)
        (2*2*4)   (2*2*9)   (3*4*4)     (3*3*6) (2*4*8)       (3*6*6)
        (2*2*2*2) (2*3*6)   (2*2*12)            (4*4*4)       (2*2*27)
                  (3*3*4)   (2*2*2*2*3)         (2*2*16)      (2*3*18)
                  (2*2*3*3)                     (2*2*4*4)     (3*3*12)
                                                (2*2*2*2*4)   (2*2*3*3*3)
                                                (2*2*2*2*2*2)

Mathematica

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[Select[facs[n], IntegerQ@*altprod]], {n, 100}]

Prog

(PARI) A347437(n, m=n, ap=1, e=0) = if(1==n, if(e%2, 1==denominator(ap), 1==numerator(ap)), sumdiv(n, d, if((d>1)&&(d<=m), A347437(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023

Crossrefs

Positions of 1's are A005117, complement A013929.

Allowing any alternating product <= 1 gives A339846.

Allowing any alternating product > 1 gives A339890.

The restriction to powers of 2 is A344607.

The even-length case is A347438, also the case of alternating product 1.

The reciprocal version is A347439.

Allowing any alternating product < 1 gives A347440.

The odd-length case is A347441.

The reverse version is A347442.

The additive version is A347446, ranked by A347457.

Allowing any alternating product >= 1 gives A347456.

The restriction to perfect squares is A347458, reciprocal A347459.

The ordered version is A347463.

A001055 counts factorizations.

A046099 counts factorizations with no alternating permutations.

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

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

A347460 counts possible alternating products of factorizations.

Cf. A025047, A038548, A062312, A088218, A119620, A316523, A330972, A332269, A347445, A347447, A347451, A347454.

Sequence in context: A368781 A050377 A344417 * A255231 A363265 A347456

Adjacent sequences: A347434 A347435 A347436 * A347438 A347439 A347440

Keyword

nonn

Author

Gus Wiseman, Sep 06 2021

Extensions

Data section extended up to a(108) by Antti Karttunen, Oct 22 2023

Status

approved