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A283183 Number of partitions of n into a prime and a square of an arbitrary integer. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A224381 A190704 A190698 * A327467 A077427 A107641

Adjacent sequences:  A283180 A283181 A283182 * A283184 A283185 A283186

KEYWORD

nonn,easy

AUTHOR

Anton Mosunov, Mar 02 2017

STATUS

approved

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Last modified May 29 03:06 EDT 2020. Contains 334696 sequences. (Running on oeis4.)