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%I #27 Jul 16 2019 11:44:31
%S 10,34,58,85,91,130,214,226,370,526,706,730,771,1255,1351,1414,1906,
%T 2986,3676,9634,21679
%N Neither square nor square + prime.
%C 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)
%C 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.
%C Hongze Li showed that there are at most O(n^0.982) members of this sequence below n, improving on earlier results of Wang.
%C a(22) > 10^11, if it exists. - _Giovanni Resta_, Jul 16 2019
%H H. Davenport, H. Heilbronn, <a href="http://plms.oxfordjournals.org/content/s2-43/1/142.extract">Note on a result in the additive theory of numbers</a>, Proceedings of the London Mathematical Society 43 (1937), pp. 142-151, DOI:<a href="https://doi.org/10.1112/plms/s2-43.2.142">10.1112/plms/s2-43.2.142</a>.
%H G. H. Hardy, J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes</a>, Acta Mathematica 44 (1923), pp. 1-70.
%H Hongze Li, <a href="https://doi.org/10.1023/A:1024513613734">The exceptional set for the sum of a prime and a square</a>, Acta Mathematica Hungarica, Vol. 99, No. 1-2 (2003), pp. 123-141.
%H R. J. Miech, <a href="https://doi.org/10.1090/S0002-9947-1968-0266873-7">On the equation n = p + x^2</a>, Transactions of the American Mathematical Society 130 (1968), pp. 494-512.
%H Yuta Suzuki, <a href="https://arxiv.org/abs/1504.04711">A remark on the conditional estimate for the sum of a prime and a square</a>, arXiv:1504.04711 [math.NT], 2015.
%H Wang Tianze, <a href="https://doi.org/10.1007/BF02274058">On the exceptional set for the equation n = p + k^2</a>, Acta Mathematica Sinica, Vol. 11, No. 2 (1995), pp. 156-167.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%t 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 *)
%o (PARI) isA020495(n)=if(issquare(n),return(0));for(k=0,sqrtint(n),if(isprime(n-k^2),return(0)));1
%K nonn,hard,more
%O 1,1
%A _David W. Wilson_
%E Comments, references, links and program from _Charles R Greathouse IV_, Aug 10 2009
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