1,1

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.

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.

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

(PARI) isA020495(n)=if(issquare(n), return(0)); for(k=0, sqrtint(n), if(isprime(n-k^2), return(0))); 1

Sequence in context: A045087 A119086 A002601 this_sequence A008527 A007584 A009924

Adjacent sequences: A020492 A020493 A020494 this_sequence A020496 A020497 A020498

nonn

David W. Wilson (davidwwilson(AT)comcast.net)

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