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

1,1

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.

Links

Table of n, a(n) for n=1..41.

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

Crossrefs

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

Sequence in context: A355901 A155990 A109556 * A277284 A090952 A347078

Adjacent sequences: A167672 A167673 A167674 * A167676 A167677 A167678

Keyword

nonn

Author

T. D. Noe, Nov 09 2009

Status

approved