A064233 Numbers that are not the sum of a prime number and a nonzero square.

1, 2, 5, 10, 13, 25, 31, 34, 37, 58, 61, 64, 85, 91, 121, 127, 130, 169, 196, 214, 226, 289, 324, 370, 379, 400, 439, 526, 529, 571, 625, 676, 706, 730, 771, 784, 829, 841, 991, 1024, 1089, 1225, 1255, 1351, 1414, 1444, 1521, 1549, 1681, 1849, 1906, 1936, 2116

Offset: 1

Author

Axel Harvey, Sep 22 2001

Keywords

• nonn
• nice

Comments

The sequence is infinite, cf. A014090. Subsequence of squares = A053726^2. Subsequence of nonsquares is disjoint union of A020495 and A065377 and so is probably finite. - Vladeta Jovovic, Apr 02 2005

Examples

			5 = 1+4 or 2+3; a prime and a square do not appear together in either sum.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• I. V. Poljakov, On the exceptional set for the sum of a prime and a perfect square, 1982 Math. USSR Izv. 19 611.

Crossrefs

Complement of A058654.

Cf. A014090, A053726, A020495, A065377.

Programs

Mathematica

Complement[ Table[ n, {n, 1, 10000} ], Union[ Flatten[ Table[ Prime[ i ] + j^2, {i, 1, 1230}, {j, 1, 100} ] ] ] ]
nspQ[n_]:=Length[Select[IntegerPartitions[n,{2}],(PrimeQ[#[[1]]] && IntegerQ[ Sqrt[ #[[2]]]])||(PrimeQ[#[[2]]]&&IntegerQ[Sqrt[#[[1]]]])&]] == 0; Select[ Range[ 2200],nspQ] (* Harvey P. Dale, Jun 18 2021 *)

PARI

list(lim)=my(v=vectorsmall(lim\1,i,1),u=List(),b);forprime(p=2,#v, b=0; while((t=p+b++^2)<=#v,v[t]=0));for(i=1,#v,if(v[i],listput(u,i))); Vec(u) \\ Charles R Greathouse IV, May 29 2012

Extensions

More terms from Vladeta Jovovic, Robert G. Wilson v and Felice Russo, Sep 23 2001