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A020495
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Neither square nor square + prime.
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2
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10, 34, 58, 85, 91, 130, 214, 226, 370, 526, 706, 730, 771, 1255, 1351, 1414, 1906, 2986, 3676, 9634, 21679
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Hardy & Littlewood's Conjecture H is that this sequence is finite and that the number of representations of n as the sum of a prime and a square is asymptotically sqrt(n)/log n * prod_{p > 2} 1 - (n / p) / (p - 1), where (n / p) is the Legendre symbol.
Hongze Li showed that there are at most O(n^0.982) members of this sequence below n, improving on earlier results of Wang.
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REFERENCES
| H. Davenport, H. Heilbronn. "Note on a result in the additive theory of numbers". Proceedings of the London Mathematical Society 43 (1937), pp. 142-151.
G. H. Hardy, J. E. Littlewood. "Some of the problems of partitio numerorum III: On the expression of a large number as a sum of primes". Acta Mathematica 44 (1923), pp. 1-70.
Hongze Li, "The exceptional set for the sum of a prime and a square". Acta Mathematica Hungarica 99 (2003), pp. 123-141.
R. J. Miech. "On the equation n = p + x^2". Transactions of the American Mathematical Society 130 (1968), pp. 494-512.
Wang Tianze, "On the exceptional set for the equation n = p + k^2". Acta Mathematica Sinica 11 (1995), pp. 156-167.
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LINKS
| Hongze Li, "The exceptional set for the sum of a prime and a square". Acta Mathematica Hungarica 99 (2003), pp. 123-141.
Eric Weisstein's World of Mathematics, Square Number
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MATHEMATICA
| isA020495[n_] := (r = True; If[ IntegerQ[ Sqrt[n]], r = False, Do[ If[ PrimeQ[n - k^2], r = False; Break[]], {k, 0, Sqrt[n]}]; r]); Select[ Range[30000], isA020495] (* From Jean-François Alcover, Oct 06 2011, after PARI *)
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PROG
| (PARI) isA020495(n)=if(issquare(n), return(0)); for(k=0, sqrtint(n), if(isprime(n-k^2), return(0))); 1
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CROSSREFS
| Sequence in context: A119086 A195900 A002601 * A155486 A008527 A007584
Adjacent sequences: A020492 A020493 A020494 * A020496 A020497 A020498
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KEYWORD
| nonn
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AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
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EXTENSIONS
| Almost certainly finite; no other terms below 25000000. Search extended to 3000000000 by James Van Buskirk without finding any more terms. - John Robertson (Jpr2718(AT)aol.com)
Comments, references, links and program from Charles R Greathouse IV Aug 10 2009
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