A167675 Least prime p such that p-2 has n divisors, or 0 if no such prime exists.

3, 5, 11, 17, 83, 47, 0, 107, 227, 569, 59051, 317, 0, 9479, 2027, 947, 0, 2207, 0, 2837, 88211, 295247, 0, 3467, 50627, 9034499, 11027, 47387, 0, 14177, 0, 15017, 1476227, 215233607, 455627, 17327, 150094635296999123, 15884240051, 89813531, 36857, 0

Offset: 1

Author

T. D. Noe, Nov 09 2009

Keywords

• nonn

Comments

This sequence is the idea of Alonso Del Arte. For n>2, a(n) is conjectured to be the smallest number that is orderly (see A167408) for n-1 values of k. For example, 11 is orderly for k=3 and 9. See A056899 for other primes p that are orderly for two k. It is a conjecture because it is not known whether there are composite numbers that are orderly for more than one value of k.

The terms a(n) for prime n are 0 except when 3^(n-1)+2 is prime. Using A051783, we find the exceptional primes to be n=2, 3, 5, 11, 37, 127, 6959.... For these n, a(n) = 3^(n-1)+2. For any n, it is easy to use the factorization of n to find the forms of numbers that have n divisors. For example, for n=38=2*19, we know that the prime must have the form 2+q*r^18 with q and r prime. The smallest such prime is 2+41*3^18.

Crossrefs

Cf. A066814 (smallest prime p such that p-1 has n divisors)

Programs

Mathematica

nn=25; t=Table[0,{nn}]; Do[p=Prime[n]; k=DivisorSigma[0,p-2]; If[k<=nn && t[[k]]==0, t[[k]]=p], {n,2,10^6}]; t