A347442 Number of factorizations of n with integer reverse-alternating product.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 3, 2, 1, 3, 3, 1, 1, 1, 7, 1, 1, 1, 8, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 8, 2, 3, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 11, 1, 1, 3, 3, 1, 1, 1, 8, 5, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 9, 1, 3, 3, 8, 1, 1, 1, 3, 1, 1, 1, 12

Offset: 1

Author

Gus Wiseman, Sep 08 2021

Keywords

• nonn

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)). The reverse-alternating product is the alternating product of the reversed sequence.

Examples

			The a(n) factorizations for n = 4, 8, 16, 32, 36, 54, 64:
  (4)    (8)      (16)       (32)         (36)       (54)     (64)
  (2*2)  (2*4)    (2*8)      (4*8)        (6*6)      (3*18)   (8*8)
         (2*2*2)  (4*4)      (2*16)       (2*18)     (2*3*9)  (2*32)
                  (2*2*4)    (2*2*8)      (3*12)     (3*3*6)  (4*16)
                  (2*2*2*2)  (2*4*4)      (2*2*9)             (2*4*8)
                             (2*2*2*4)    (2*3*6)             (4*4*4)
                             (2*2*2*2*2)  (3*3*4)             (2*2*16)
                                          (2*2*3*3)           (2*2*2*8)
                                                              (2*2*4*4)
                                                              (2*2*2*2*4)
                                                              (2*2*2*2*2*2)
		

Links

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

Crossrefs

The restriction to powers of 2 is A000041, reverse A344607.

Positions of 2's are A001248.

Positions of 1's are A005117.

Positions of non-1's are A013929.

Allowing any alternating product <= 1 gives A339846.

Allowing any alternating product > 1 gives A339890.

The non-reverse version is A347437.

The reciprocal version is A347438.

The even-length case is A347439.

Allowing any alternating product < 1 gives A347440.

The odd-length case is A347441, ranked by A347453.

The additive version is A347445, ranked by A347457.

The non-reverse additive version is A347446, ranked by A347454.

Allowing any alternating product >= 1 gives A347456.

The ordered version is A347463.

A038548 counts possible reverse-alternating products of factorizations.

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

A236913 counts partitions of 2n with reverse-alternating sum <= 0.

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

Cf. A000290, A025047, A330972, A347443, A347449, A347451, A347458, A347459, A347460, A347462.

Programs

Mathematica

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

PARI

A347442(n, m=n, ap=1, e=0) = if(1==n, 1==denominator(ap), sumdiv(n, d, if((d>1)&&(d<=m), A347442(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023

Formula

a(2^n) = A000041(n).

Extensions

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