

A167675


Least prime p such that p2 has n divisors, or 0 if no such prime exists.


0



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 n1 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^(n1)+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^(n1)+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



MATHEMATICA

nn=25; t=Table[0, {nn}]; Do[p=Prime[n]; k=DivisorSigma[0, p2]; If[k<=nn && t[[k]]==0, t[[k]]=p], {n, 2, 10^6}]; t


CROSSREFS

Cf. A066814 (smallest prime p such that p1 has n divisors)


KEYWORD

nonn


AUTHOR



STATUS

approved



