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 A322953 Number of stable modules [f, g + sqrt(n)] in canonical form. 0
 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 (list; graph; refs; listen; history; text; internal format)
 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 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: A332845 A190621 A325314 * A049268 A291305 A004179 Adjacent sequences:  A322950 A322951 A322952 * A322954 A322955 A322956 KEYWORD nonn,changed AUTHOR Eric Rowland, Dec 31 2018 STATUS approved

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Last modified May 12 05:30 EDT 2021. Contains 343812 sequences. (Running on oeis4.)