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A020495
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Neither square nor square + prime.
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6
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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
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OFFSET
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1,1
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COMMENTS
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Almost certainly finite; no other terms below 2.5*10^7. Search extended to 3*10^9 by James Van Buskirk without finding any more terms. - John Robertson (Jpr2718(AT)aol.com)
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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LINKS
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MATHEMATICA
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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] (* Jean-François Alcover, Oct 06 2011, after PARI *)
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PROG
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(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
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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