cp's OEIS Frontend

A074386 Numbers k such that sigma(k) is the square of a prime.

%I A074386 #37 Dec 13 2024 12:39:35
%S A074386 3,81,400
%N A074386 Numbers k such that sigma(k) is the square of a prime.
%C A074386 The next term, if it exists, is > 10^11. - _Donovan Johnson_, Aug 24 2012
%C A074386 a(4), if it exists, satisfies sigma(a(4)) > 10^36.  - _Hiroaki Yamanouchi_, Sep 10 2014
%C A074386 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 A074386 M. A. Bennett and A. Levin, <a href="https://doi.org/10.1007/s00605-015-0748-1">The Nagell-Ljunggren equation via Runge’s method</a>, Monatshefte für Mathematik 177:1 (2015), 15-31.
%H A074386 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goormaghtigh_conjecture">Goormaghtigh conjecture</a>
%e A074386 sigma[{3,81,400}]={4,121,961}.
%t A074386 Do[s=DivisorSigma[1, n]; If[PrimeQ[Sqrt[s]], Print[n]], {n, 1, 1000000}] (* Corrected by _N. J. A. Sloane_, May 26 2008 *)
%Y A074386 Cf. A000203, A001248, A008848, A023194, A028982.
%Y A074386 Cf. A084738, A065854, A128164.
%Y A074386 Subsequence of A006532.
%K A074386 nonn,bref,more
%O A074386 1,1
%A A074386 _Labos Elemer_, Aug 22 2002
%E A074386 Definition corrected by Juan Lopez, May 26 2008
%E A074386 Edited by _N. J. A. Sloane_, May 26 2008