A166853 a(n) is the smallest number m such that m^m-n is prime, or zero if there is no such m.

2, 2, 8, 3, 4, 5, 6, 3, 0, 3, 78, 13, 6, 3, 4, 3, 4, 17, 12, 3, 118, 3, 4, 3, 3

Offset: 1

Author

Farideh Firoozbakht and M. F. Hasler, Nov 27 2009

Keywords

• hard
• more
• nonn

Comments

The sequence with the unknown terms a(n) indicated by -n:

(0's occur for n=9, 49, 81, 121....)

2,2,8,3,4,5,6,3,0,3,78,13,6,3,4,3,4,17,12,3,118,3,4,3,3,

-26,4,-28,4,487,90,9,4,-34,24,5,6,271,28,969,-41,5,-43,7,4,5,32,37,0,621,

20,15,34,7,6,9,4,5,4,7,-61,7,4,5,4,-66,6,63,134,27,10,35,102,31,4,

5,4,569,-79,13,0,15,4,5,-85,7,110,5,4,131,1122,7,4,11,8,7,6,9,4,-100,

22,5,-103,-104,4,5,4,11,12,39,-111,...

If they exist, the first two unknown terms, a(26) and a(28), they are greater than 10000. All other unknown terms a(n), for n<112 are greater than 4000.

If it exists, a(26) > 25000. - Robert Price, Apr 26 2019

Examples

			We have a(1)=2 since 1^1-1 is not prime, but 2^2-1 is prime.
a(9)=0 since 2^2-9 is not prime, and if m is an even number greater than 2 then m^m-9=(m^(m/2)-3)*(m^(m/2)+3) is composite. So there is no number m such that m^m-9 is prime. The same applies to any odd square > 25.
We have a(25)=3 since 3^3-25=2 is prime. But 25 is the only known square of the form m^m-2, so a(n)=0 for other odd squares > 25, e.g., n = 49,81,121,....
a(115)=2736 is the largest known term. 2736^2736-115 is a probable prime.
		

Crossrefs

Cf. A087037, A087038, A016754, A016923, A100407, A100408, A166852.

Formula

a(n)=0 if n=3^2 or n=(2k+1)^2 > 25, or n = (6k+1)^3 = A016923(k) with k>0.