

A273013


Number of different arrangements of nonnegative integers on a pair of nsided dice such that the dice can add to any integer from 0 to n^21.


20



1, 1, 1, 3, 1, 7, 1, 10, 3, 7, 1, 42, 1, 7, 7, 35, 1, 42, 1, 42, 7, 7, 1, 230, 3, 7, 10, 42, 1, 115, 1, 126, 7, 7, 7, 393, 1, 7, 7, 230, 1, 115, 1, 42, 42, 7, 1, 1190, 3, 42, 7, 42, 1, 230, 7, 230, 7, 7, 1, 1158, 1, 7, 42, 462, 7, 115, 1, 42, 7, 115, 1, 3030
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OFFSET

1,4


COMMENTS

The set of b values (see formula), and therefore also a(n), depends only on the prime signature of n. So, for example, a(24) will be identical to a(n) of any other n which is also of the form p_1^3*p_2, (e.g., 40, 54, 56).
The value of b_1 will always be 1. When n is prime, the only nonzero b will be b_1, so therefore a(n) will be 1.
In any arrangement, both dice will have a 0, and one will have a 1 (here called the lead die). To determine any one of the actual arrangements to numbers on the dice, select one of the permutations of divisors (for the lead die), then select another permutation that is either the same length as that of the lead die, or one less. For example, if n = 12, we might select 2*3*2 for the lead die, and 3*4 for the second die. These numbers effectively tell you when to "switch track" when numbering the dice, and will uniquely result in the numbering: (0,1,6,7,12,13,72,73,78,79,84,85; 0,2,4,18,20,22,36,38,40,54,56,58).
a(n) is the number of (unordered) pairs of polynomials c(x) = x^c_1 + x^c_2 + ... + x^c_n, d(x) = x^d_1 + x^d_2 + ... + x^d_n with nonnegative integer exponents such that c(x)*d(x) = (x^(n^2)1)/(x1).  Alois P. Heinz, May 13 2016
a(n) is also the number of principal reversible squares of order n.  S. Harry White, May 19 2018
Also the number of ordered factorizations of n^2 with alternating product 1. This follows from the author's formula. Taking n instead of n^2 gives a(sqrt(n)) if n is a perfect square, otherwise 0. Here, an ordered factorization of n is a sequence of positive integers > 1 with product n, and the alternating product of a sequence (y_1,...,y_k) is Product_i y_i^((1)^(i1)). For example, the a(1) = 1 through a(9) = 3 factorizations are:
() (22) (33) (44) (55) (66) (77) (88) (99)
(242) (263) (284) (393)
(2222) (362) (482) (3333)
(2233) (2244)
(2332) (2442)
(3223) (4224)
(3322) (4422)
(22242)
(24222)
(222222)
(End)


LINKS



FORMULA

a(n) = b_1^2 + b_2^2 + b_3^2 + ... + b_1*b_2 + b_2*b_3 + b_3*b_4 + ..., where b_k is the number of different permutations of k divisors of n to achieve a product of n. For example, for n=24, b_3 = 9 (6 permutations of 2*3*4 and 3 permutations of 2*2*6).
a(n) = r(n,n) where r(m,1) = 1 and r(m,n) = Sum_{dm,d<m} r(n,d) if n != 1.  William P. Orrick, Feb 19 2023


EXAMPLE

When n = 4, a(n) = 3; the three arrangements are (0,1,2,3; 0,4,8,12), (0,1,4,5; 0,2,8,10), (0,1,8,9; 0,2,4,6).
When n = 5, a(n) = 1; the sole arrangement is (0,1,2,3,4; 0,5,10,15,20).


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)^(i1), {i, Length[q]}];
Table[Length[Select[Join@@Permutations/@facs[n^2], altprod[#] == 1&]], {n, 30}]
(* or *)
ofc[n_, k_] := If[k > PrimeOmega[n], 0, If[k == 0 && n == 1, 1, Sum[ofc[n/d, k1], {d, Rest[Divisors[n]]}]]];
Table[If[n == 1, 1, Sum[ofc[n, x]^2 + ofc[n, x]*ofc[n, x+1], {x, n}]], {n, 30}]
(* Gus Wiseman, Oct 29 2021, based on author's formula *)


PROG

(PARI)
A273013aux(n, k=0, t=1) = if(1==n, (1==t), my(s=0); fordiv(n, d, if((d>1), s += A273013aux(n/d, 1k, t*(d^((1)^k))))); (s));
(SageMath)
@cached_function
def r(m, n):
if n==1:
return(1)
divList = divisors(m)[:1]
return(sum(r(n, d) for d in divList))


CROSSREFS

Positions of 1's are 1 and A000040.
A119620 counts partitions with alternating product 1, ranked by A028982.
A347438 counts factorizations with alternating product 1.
A347460 counts possible alternating products of factorizations.
A347463 counts ordered factorizations with integer alternating product.
A347466 counts factorizations of n^2.


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



