A283183 Number of partitions of n into a prime and a square of an arbitrary integer.

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, 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, 4, 2, 5, 6, 3, 2, 6, 2, 8, 4, 3, 6, 2, 2, 11, 6, 0, 6, 6

Offset: 1

Author

Anton Mosunov, Mar 02 2017

Keywords

• nonn
• easy

Comments

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.

a(n) is odd if and only if n is prime.

Examples

			a(11) = 5 because 11 = 11 + 0^2 = 7 + (-2)^2 = 7 + 2^2 = 2 + (-3)^2 = 2 + 3^2.
		

Links

• Anton Mosunov, Table of n, a(n) for n = 1..10000
• H. Li, The exceptional set for the sum of a prime and a square, Acta Math. Hung., 99:123 (2003), 123-141.
• R. J. Miech, On the equation n=p+x^2, Trans. of the AMS, 130:3 (1968), 494-512.
• A. Nayebi, Upper bounds on the solutions to n=p+m^2, Bull of the Iran. Math. Soc., 37:4 (2011), 95-108.
• W. Tianze, On the exceptional set for the equation n=p+k^2, Acta Math. Sinica, 11:2 (1995), 156-167.

Crossrefs

Cf. A002471, A020495.

Programs

Mathematica

a[n_] := Boole@ PrimeQ[n] + 2 Length@ Select[n - Range[Sqrt@ n]^2, PrimeQ]; Array[a, 87] (* Giovanni Resta, Apr 09 2017 *)

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