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A078116
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Primes of the form x^2 + 2y^2 where y<=x. Terms are listed in increasing order of x; for fixed x they're in increasing order of y.
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0
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3, 11, 17, 43, 67, 83, 89, 113, 131, 179, 139, 193, 283, 241, 331, 457, 227, 233, 257, 353, 467, 563, 617, 307, 577, 739, 379, 433, 523, 811, 1009, 443, 449, 491, 569, 641, 683, 953, 1019, 1163, 547, 601, 691, 643, 787, 1777, 761, 827, 857, 929, 971, 1307
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Every prime of the form 8n+1 or 8n+3 has a unique representation of the form x^2 + 2y^2 with positive integers x and y. This sequence has the primes for which y<=x.
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REFERENCES
| Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford University Press 1972, p. 276 (Fermat prime number theorems).
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MATHEMATICA
| Select[Flatten[Table[x^2+2y^2, {x, 0, 30}, {y, 0, x}]], PrimeQ]
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PROG
| (PARI) sqplus2sq(n, m) = ct=0; for(x=1, n, for(y=1, x, s = x^2+m*y^2; if(isprime(s), ct+=1; print1(s" "); ); ); ); \\ Lists primes of the form x^2+m*y^2 with 1<=y<=x<=n.
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CROSSREFS
| Sequence in context: A024592 A203162 A176804 * A127996 A032008 A061368
Adjacent sequences: A078113 A078114 A078115 * A078117 A078118 A078119
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KEYWORD
| easy,nonn
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AUTHOR
| Cino Hilliard (hillcino368(AT)gmail.com), Dec 05 2002
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EXTENSIONS
| Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 12 2002
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