A105016 Smallest a(n) such that a(n)^2 - n is a positive prime, or 0 if no such a(n) exists.

0, 2, 2, 4, 3, 4, 3, 3, 5, 4, 9, 4, 5, 4, 4, 14, 0, 6, 5, 6, 5, 8, 5, 5, 11, 6, 7, 8, 9, 6, 7, 6, 7, 6, 6, 8, 7, 12, 7, 10, 9, 8, 7, 12, 7, 8, 7, 7, 11, 0, 9, 8, 9, 8, 11, 12, 13, 8, 9, 8, 11, 8, 8, 10, 9, 12, 13, 18, 9, 10, 9, 10, 13, 12, 9, 16, 9, 10, 9, 9, 11, 10, 21, 10, 11, 12, 13, 10, 15, 10

Offset: 0

Author

Joshua Zucker, Mar 31 2005

Keywords

• nonn

Comments

An old ARML problem asked for the smallest n>0 such that a(n) does not exist.

Examples

			a(8) = 5 because 5^2 - 8 = 17 is the smallest square that gives a prime difference.
a(16) = 0 because if x^2 - 16 is prime, then a prime equals (x+4)(x-4), which is impossible.
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A075555 for the primes = a(n)^2 - n.

Programs

Mathematica

Table[s = Sqrt[n]; If[IntegerQ[s], If[PrimeQ[(s + 1)^2 - n], k = s + 1, k = 0], k = Ceiling[s]; While[! PrimeQ[k^2 - n], k++]]; k, {n, 0, 100}] (* T. D. Noe, Apr 17 2011 *)