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

3, 81, 400

Offset: 1

Author

Labos Elemer, Aug 22 2002

Keywords

• nonn
• bref
• more

Comments

The next term, if it exists, is > 10^11. - Donovan Johnson, Aug 24 2012

a(4), if it exists, satisfies sigma(a(4)) > 10^36. - Hiroaki Yamanouchi, Sep 10 2014

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

Examples

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

Links

• M. A. Bennett and A. Levin, The Nagell-Ljunggren equation via Runge’s method, Monatshefte für Mathematik 177:1 (2015), 15-31.
• Wikipedia, Goormaghtigh conjecture

Crossrefs

Cf. A000203, A001248, A008848, A023194, A028982.

Cf. A084738, A065854, A128164.

Subsequence of A006532.

Programs

Mathematica

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

Extensions

Definition corrected by Juan Lopez, May 26 2008

Edited by N. J. A. Sloane, May 26 2008