Search: id:A020495
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%I A020495
%S A020495 10,34,58,85,91,130,214,226,370,526,706,730,771,1255,1351,1414,1906,2986,
%T A020495 3676,9634,21679
%N A020495 Neither square nor square + prime.
%C A020495 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 A020495 Hongze Li showed that there are at most O(n^0.982) members of this sequence 
               below n, improving on earlier results of Wang.
%D A020495 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.
%D A020495 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.
%D A020495 Hongze Li, "The exceptional set for the sum of a prime and a square". 
               Acta Mathematica Hungarica 99 (2003), pp. 123-141.
%D A020495 R. J. Miech. "On the equation n = p + x^2". Transactions of the American 
               Mathematical Society 130 (1968), pp. 494-512.
%D A020495 Wang Tianze, "On the exceptional set for the equation n = p + k^2". Acta 
               Mathematica Sinica 11 (1995), pp. 156-167.
%H A020495 Hongze Li, "<a href="http://math.sjtu.edu.cn/teacher/lihz/list.htm">The 
               exceptional set for the sum of a prime and a square</a>". Acta Mathematica 
               Hungarica 99 (2003), pp. 123-141.
%H A020495 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SquareNumber.html">Square Number</a>
%o A020495 (PARI) isA020495(n)=if(issquare(n),return(0));for(k=0,sqrtint(n),if(isprime(n-k^2),
               return(0)));1
%Y A020495 Sequence in context: A045087 A119086 A002601 this_sequence A008527 A007584 
               A009924
%Y A020495 Adjacent sequences: A020492 A020493 A020494 this_sequence A020496 A020497 
               A020498
%K A020495 nonn
%O A020495 1,1
%A A020495 David W. Wilson (davidwwilson(AT)comcast.net)
%E A020495 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)
%E A020495 Comments, references, links and program from Charles R Greathouse IV 
               Aug 10 2009

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