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

1,3

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.

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.

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A002471, A020495.

Sequence in context: A190704 A346611 A190698 * A327467 A347832 A077427

Adjacent sequences: A283180 A283181 A283182 * A283184 A283185 A283186

Keyword

nonn,easy

Author

Anton Mosunov, Mar 02 2017

Status

approved