A322953 Number of stable modules [f, g + sqrt(n)] in canonical form.

0, 2, 3, 0, 5, 6, 7, 7, 0, 10, 9, 11, 13, 10, 12, 0, 13, 14, 17, 15, 18, 18, 13, 20, 0, 16, 18, 25, 21, 20, 25, 20, 24, 26, 18, 0, 31, 18, 26, 34, 27, 24, 29, 29, 31, 34, 19, 31, 0, 24, 32, 39, 29, 32, 36, 34, 38, 36, 27, 40, 47, 22, 31, 0, 38, 36, 41, 33, 44

Offset

1,2

Comments

Stable modules arise in Edwards's recasting of Gauss's theory of binary quadratic forms. See either Edwards reference for the definitions of stable modules and canonical form.

References

Harold M. Edwards, Essays in Constructive Mathematics, Springer, 2005, page 80 and pages 90-92.

Harold M. Edwards, Higher Arithmetic: An Algorithmic Introduction to Number Theory, American Mathematical Society, 2008, page 119 and pages 169-177.

Links

Table of n, a(n) for n=1..69.

Formula

a(n) = 0 when n is a square.

Example

For n = 5 the a(5) = 5 stable modules are [1, sqrt(5)], [5, sqrt(5)], [2, 1 + sqrt(5)], [4, 1 + sqrt(5)], [4, 3 + sqrt(5)].  Applying Edwards' comparison algorithm to each stable module partitions them into two cycles:  [1, sqrt(5)] -> [4, 3 + sqrt(5)] -> [5, sqrt(5)] -> [4, 1 + sqrt(5)] -> [1, sqrt(5)] and [2, 1 + sqrt(5)] -> [2, 1 + sqrt(5)].

Mathematica

Table[
  Length[If[
    IntegerQ[Sqrt[n]],
    {},
    Join @@ Table[
      Join @@ Function[f,
        If[k == 0 || 2 k == f,
          {{f, k + Sqrt[n]}},
          {{f, k + Sqrt[n]}, {f, f - k + Sqrt[n]}}
        ]
      ] /@ Select[Divisors[n - k^2], Function[f, f >= 2 k]],
      {k, 0, Sqrt[n]}
    ]
  ]],
  {n, 1, 100}
]

Crossrefs

Sequence in context: A351079 A190621 A325314 * A049268 A291305 A004179

Adjacent sequences: A322950 A322951 A322952 * A322954 A322955 A322956

Keyword

nonn

Author

Eric Rowland, Dec 31 2018

Status

approved