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A020669
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Numbers of form x^2 + 5 y^2.
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20
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0, 1, 4, 5, 6, 9, 14, 16, 20, 21, 24, 25, 29, 30, 36, 41, 45, 46, 49, 54, 56, 61, 64, 69, 70, 80, 81, 84, 86, 89, 94, 96, 100, 101, 105, 109, 116, 120, 121, 125, 126, 129, 134, 141, 144, 145, 149, 150, 161, 164, 166, 169, 174, 180, 181, 184, 189, 196, 201, 205, 206, 214, 216
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OFFSET
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1,3
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COMMENTS
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In other words, numbers represented by quadratic form with Gram matrix [1,0; 0,5].
x^2 + 5 y^2 has discriminant -20.
A positive integer n is in this sequence if and only if the p-adic order ord_p(n) of n is even for any prime p with floor(p/10) odd, and the number of prime divisors p == 3 or 7 (mod 20) of n with ord_p(n) odd has the same parity with ord_2(n). - Zhi-Wei Sun, Mar 24 2018
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REFERENCES
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H. Cohn, A second course in number theory, John Wiley & Sons, Inc., New York-London, 1962. See pp. 3, 4 and later chapters.
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989. See Eq. (2.22), p. 33.
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LINKS
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FORMULA
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MAPLE
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select(t -> [isolve(x^2+5*y^2=t)]<>[], [$0..1000]); # Robert Israel, May 11 2016
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MATHEMATICA
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formQ[n_] := Reduce[x >= 0 && y >= 0 && n == x^2 + 5 y^2, {x, y}, Integers] =!= False; Select[ Range[0, 300], formQ] (* Jean-François Alcover, Sep 20 2011 *)
mx = 300;
limx = Sqrt[mx]; limy = Sqrt[mx/5];
Select[
Union[
Flatten[
Table[x^2 + 5*y^2, {x, 0, limx}, {y, 0, limy}]
]
], # <= mx &
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PROG
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CROSSREFS
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For the properly represented numbers see A344231.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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