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%I #30 Jun 25 2023 03:32:58
%S 1,2,5,10,13,25,31,34,37,58,61,64,85,91,121,127,130,169,196,214,226,
%T 289,324,370,379,400,439,526,529,571,625,676,706,730,771,784,829,841,
%U 991,1024,1089,1225,1255,1351,1414,1444,1521,1549,1681,1849,1906,1936,2116
%N Numbers that are not the sum of a prime number and a nonzero square.
%C 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
%H T. D. Noe, <a href="/A064233/b064233.txt">Table of n, a(n) for n = 1..1000</a>
%H I. V. Poljakov, <a href="http://dx.doi.org/10.1070/IM1982v019n03ABEH001429">On the exceptional set for the sum of a prime and a perfect square</a>, 1982 Math. USSR Izv. 19 611.
%e 5 = 1+4 or 2+3; a prime and a square do not appear together in either sum.
%t Complement[ Table[ n, {n, 1, 10000} ], Union[ Flatten[ Table[ Prime[ i ] + j^2, {i, 1, 1230}, {j, 1, 100} ] ] ] ]
%t 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 *)
%o (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
%Y Complement of A058654.
%Y Cf. A014090, A053726, A020495, A065377.
%K nonn,nice
%O 1,2
%A _Axel Harvey_, Sep 22 2001
%E More terms from _Vladeta Jovovic_, _Robert G. Wilson v_ and _Felice Russo_, Sep 23 2001
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