A020495 Neither square nor square + prime.

10, 34, 58, 85, 91, 130, 214, 226, 370, 526, 706, 730, 771, 1255, 1351, 1414, 1906, 2986, 3676, 9634, 21679

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• hard
• more

Comments

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.

a(22) > 10^11, if it exists. - Giovanni Resta, Jul 16 2019

Links

• 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, DOI:10.1112/plms/s2-43.2.142.
• G. H. Hardy, J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a 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, Vol. 99, No. 1-2 (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.
• Yuta Suzuki, A remark on the conditional estimate for the sum of a prime and a square, arXiv:1504.04711 [math.NT], 2015.
• Wang Tianze, On the exceptional set for the equation n = p + k^2, Acta Mathematica Sinica, Vol. 11, No. 2 (1995), pp. 156-167.
• Eric Weisstein's World of Mathematics, Square Number

Programs

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] (* Jean-François Alcover, Oct 06 2011, after PARI *)

PARI

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

Extensions

Comments, references, links and program from Charles R Greathouse IV, Aug 10 2009