A347459 Number of factorizations of n^2 with integer reciprocal alternating product.

1, 1, 1, 3, 1, 4, 1, 6, 3, 4, 1, 11, 1, 4, 4, 12, 1, 11, 1, 12, 4, 4, 1, 28, 3, 4, 6, 12, 1, 19, 1, 22, 4, 4, 4, 38, 1, 4, 4, 29, 1, 21, 1, 12, 11, 4, 1, 65, 3, 11, 4, 12, 1, 29, 4, 29, 4, 4, 1, 71, 1, 4, 11, 40, 4, 22, 1, 12, 4, 18, 1, 107, 1, 4, 11, 12, 4, 22, 1, 66, 12, 4, 1, 76, 4, 4, 4, 30, 1, 71, 4, 12, 4, 4, 4, 141

Offset: 1

Author

Gus Wiseman, Sep 22 2021

Keywords

• nonn

Comments

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

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

All such factorizations have even length.

Image appears to be 1, 3, 4, 6, 11, ... , missing some numbers such as 2, 5, 7, 8, 9, ...

The case of alternating product 1, the case of alternating sum 0, and the reverse version are all counted by A001055.

Examples

			The a(2) = 1 through a(10) = 4 factorizations:
    2*2  3*3  2*8      5*5  6*6      7*7  8*8          9*9      2*50
              4*4           2*18          2*32         3*27     5*20
              2*2*2*2       3*12          4*16         3*3*3*3  10*10
                            2*2*3*3       2*2*2*8               2*2*5*5
                                          2*2*4*4
                                          2*2*2*2*2*2
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Positions of 1's are 1 and A000040, squares A001248.

The additive version (partitions) is A000041, the even bisection of A119620.

Partitions of this type are ranked by A028982 and A347451.

The restriction to powers of 2 is A236913, the even bisection of A027187.

The nonsquared nonreciprocal even-length version is A347438.

This is the restriction to perfect squares of A347439.

The nonreciprocal version is A347458, non-squared A347437.

A000290 lists squares, complement A000037.

A001055 counts factorizations.

A046099 counts factorizations with no alternating permutations.

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

A347460 counts possible alternating products of factorizations.

A339846 counts even-length factorizations.

A339890 counts odd-length factorizations.

A347457 ranks partitions with integer alternating product.

A347466 counts factorizations of n^2.

Cf. A062312, A316523, A330972, A344653, A346635, A347440, A347441, A347442, A347453, A347461, A347463, A347464, A347705.

Programs

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^2],IntegerQ[recaltprod[#]]&]],{n,100}]

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))));
A347459(n) = A347439(n^2); \\ Antti Karttunen, Jul 28 2024

Formula

a(2^n) = A236913(n).

a(n) = A347439(n^2).

Extensions

Data section extended up to a(96) by Antti Karttunen, Jul 28 2024