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

1,2

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

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.

Example

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

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 *)

Prog

(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

Crossrefs

Complement of A058654.

Cf. A014090, A053726, A020495, A065377.

Sequence in context: A135467 A230550 A018571 * A051952 A103188 A281229

Adjacent sequences: A064230 A064231 A064232 * A064234 A064235 A064236

Keyword

nonn,nice

Author

Axel Harvey, Sep 22 2001

Extensions

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

Status

approved