A109905 a(n) = greatest prime of the form k*(n-k) +1. 0 if no such prime exists.

0, 2, 3, 5, 7, 0, 13, 17, 19, 17, 31, 37, 43, 41, 37, 61, 73, 73, 89, 101, 109, 113, 131, 109, 157, 89, 181, 197, 211, 0, 241, 257, 271, 281, 307, 181, 337, 353, 379, 401, 421, 433, 463, 449, 487, 521, 547, 577, 601, 617, 631, 677, 701, 0, 757, 769, 811, 761, 859, 757

Offset: 1

Author

Amarnath Murthy, Jul 15 2005

Keywords

• nonn

Comments

k can take values from 1 to floor[n/2].

a(n)=0 for k = 1, 6, 30 and 54. Are there any others? - Robert Israel, Feb 23 2018

There are none for n up to 10^9. - Mauro Fiorentini, Jul 24 2023

Examples

			a(15) = 37 as 1*14 +1 = 16, 2*13 +1 = 27 are composite but 3*12 +1= 37 is a prime.
a(6) = 0 as 1*5 +1=6, 2*4 +1=9, 3*3 +1 = 10 are all composite.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A109904, A026728.

Programs

Maple

f:= proc(n) local k;
  for k from floor(n/2) to 1 by -1 do
    if isprime(k*(n-k)+1) then return k*(n-k)+1 fi
  od:
  0 end proc:
map(f, [$1..100]); # Robert Israel, Feb 23 2018

Mathematica

Table[Max@Prepend[Select[Table[k (n - k) + 1, {k, n/2}], PrimeQ], 0], {n, 60}] (* Ivan Neretin, Feb 23 2018 *)

PARI

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

Extensions

More terms from Max Alekseyev, Oct 04 2005