A074386 - OEIS

%I #32 Mar 15 2019 13:17:15

%S 3,81,400

%N Numbers n such that sigma(n) is the square of a prime.

%C The next term, if it exists, is > 10^11. - _Donovan Johnson_, Aug 24 2012

%C a(4), if it exists, satisfies sigma(a(4)) > 10^36.  - _Hiroaki Yamanouchi_, Sep 10 2014

%C If n belongs to this sequence, it may have at most two distinct prime divisors. If n=p^k, then sigma(p^k) = (p^(k+1)-1)/(p-1) = r^2 for some prime r. For k=1, it trivially has the only solution n=3; while for k>1 it is a partial case of the Nagell-Ljunggren equation and has the only prime solution r=11 (see Bennett-Levin 2015) corresponding to n=3^4=81. If n=p^k*q^m, then sigma(n) = (p^(k+1)-1)/(p-1) * (q^(m+1)-1)/(q-1) = r^2 for some prime r, implying that (p^(k+1)-1)/(p-1) = (q^(m+1)-1)/(q-1) = r. Here k+1 and m+1 must be odd distinct primes. The Goormaghtigh conjecture would imply that its only solution is n=400 with (p,k,q,m)=(5,2,2,4). - _Max Alekseyev_, Apr 24 2015

%H M. A. Bennett and A. Levin, <a href="http://link.springer.com/article/10.1007%2Fs00605-015-0748-1">The Nagell-Ljunggren equation via Runge’s method</a>, Monatshefte für Mathematik 177:1 (2015), 15-31.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goormaghtigh_conjecture">Goormaghtigh conjecture</a>

%e sigma[{3,81,400}]={4,121,961}.

%t Do[s=DivisorSigma[1, n]; If[PrimeQ[Sqrt[s]], Print[n]], {n, 1, 1000000}] (* Corrected by _N. J. A. Sloane_, May 26 2008 *)

%Y Cf. A000203, A001248, A023194, A028982.

%Y Cf. A084738, A065854, A128164.

%Y Subsequence of A006532.

%K nonn,bref,more

%O 1,1

%A _Labos Elemer_, Aug 22 2002

%E Definition corrected by Juan Lopez, May 26 2008

%E Edited by _N. J. A. Sloane_, May 26 2008