A026728 a(n) = number of primes of the form k*(n-k) + 1.

0, 1, 1, 1, 2, 0, 3, 2, 1, 1, 4, 1, 6, 1, 1, 2, 8, 1, 5, 3, 1, 4, 7, 1, 7, 1, 4, 5, 8, 0, 10, 6, 2, 2, 7, 1, 9, 8, 4, 4, 14, 1, 16, 3, 3, 5, 12, 3, 7, 7, 4, 11, 21, 0, 11, 4, 7, 6, 11, 2, 12, 9, 7, 10, 7, 1, 22, 7, 7, 5, 17, 3, 23, 10, 2, 9, 19, 2, 19, 8, 5, 8, 23, 1, 16, 6, 4, 11, 14, 4, 16, 12, 9, 5, 12

Offset: 1

Author

Reinhard Zumkeller, Jan 27 2004

Keywords

• nonn

Comments

Number of primes of form x*y+1 with x+y=n.

For n <= 10^7, a(n) = 0 only for n = 1, 6, 30 and 54. - Robert Israel, Oct 30 2024

Examples

			a(7) = 3, 1*6 +1 = 7, 2*5 +1 = 11, 3*4 +1 = 13.
n=16: {m: m=x*y+1 and x+y=16} = {16,29,40,49,56,61,64,65} containing two primes: 29 and 61, therefore a(16)=2.
n = 7 is the only number which gives primes for all possible values of k.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A109904, A109905.

Programs

Maple

f:= proc(n) local k;
  nops(select(isprime,[seq(k*(n-k)+1, k=1..n/2)]))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 30 2024

Mathematica

a[n_] := Select[(Times @@ # + 1&) /@ IntegerPartitions[n, {2}], PrimeQ] // Length;
Array[a, 95] (* Jean-François Alcover, Aug 02 2018 *)

PARI

{ a(n)=local(r);r=0;for(k=1,n\2,if(isprime(k*(n-k)+1),r++));r } \\ Max Alekseyev, Oct 04 2005

Extensions

More terms from Max Alekseyev, Oct 04 2005

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar