cp's OEIS Frontend

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

%I A167675 #2 Mar 30 2012 17:22:55
%S A167675 3,5,11,17,83,47,0,107,227,569,59051,317,0,9479,2027,947,0,2207,0,
%T A167675 2837,88211,295247,0,3467,50627,9034499,11027,47387,0,14177,0,15017,
%U A167675 1476227,215233607,455627,17327,150094635296999123,15884240051,89813531,36857,0
%N A167675 Least prime p such that p-2 has n divisors, or 0 if no such prime exists.
%C A167675 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.
%C A167675 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.
%t A167675 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
%Y A167675 Cf. A066814 (smallest prime p such that p-1 has n divisors)
%K A167675 nonn
%O A167675 1,1
%A A167675 _T. D. Noe_, Nov 09 2009