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%I #20 Apr 10 2017 12:44:51
%S 0,1,3,2,1,4,3,2,2,0,5,4,1,4,2,2,3,4,3,4,4,2,5,2,0,2,6,4,3,4,1,6,4,0,
%T 4,2,1,8,4,2,5,4,3,4,4,2,7,4,2,2,4,4,5,6,2,6,4,0,5,4,1,8,4,0,4,6,5,8,
%U 4,2,5,6,3,2,6,2,8,4,3,6,2,2,11,6,0,6,6
%N Number of partitions of n into a prime and a square of an arbitrary integer.
%C a(n) is also the number of solutions to the equation n = p + m^2, where p is prime and m is an arbitrary integer. In comparison, the sequence A002471 counts representations with m being nonnegative.
%C a(n) is odd if and only if n is prime.
%H Anton Mosunov, <a href="/A283183/b283183.txt">Table of n, a(n) for n = 1..10000</a>
%H H. Li, <a href="https://doi.org/10.1023/A:1024513613734">The exceptional set for the sum of a prime and a square</a>, Acta Math. Hung., 99:123 (2003), 123-141.
%H R. J. Miech, <a href="https://doi.org/10.2307/1994761">On the equation n=p+x^2</a>, Trans. of the AMS, 130:3 (1968), 494-512.
%H A. Nayebi, <a href="https://arxiv.org/abs/1004.0536">Upper bounds on the solutions to n=p+m^2</a>, Bull of the Iran. Math. Soc., 37:4 (2011), 95-108.
%H W. Tianze, <a href="https://doi.org/10.1007/BF02274058">On the exceptional set for the equation n=p+k^2</a>, Acta Math. Sinica, 11:2 (1995), 156-167.
%e a(11) = 5 because 11 = 11 + 0^2 = 7 + (-2)^2 = 7 + 2^2 = 2 + (-3)^2 = 2 + 3^2.
%t a[n_] := Boole@ PrimeQ[n] + 2 Length@ Select[n - Range[Sqrt@ n]^2, PrimeQ]; Array[a, 87] (* _Giovanni Resta_, Apr 09 2017 *)
%o (PARI) local(i,j,k,total); for (i=1, 1000, j=1; k=1; total=isprime(i); while (j <= i, total += 2*isprime(i-j); j += (2*k+1); k++); print1(total, ", ")) // _Anton Mosunov_, Apr 09 2017
%Y Cf. A002471, A020495.
%K nonn,easy
%O 1,3
%A _Anton Mosunov_, Mar 02 2017
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