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

3, 81, 400

Offset

1,1

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

Links

Table of n, a(n) for n=1..3.

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

Example

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

Mathematica

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

Crossrefs

Cf. A000203, A001248, A023194, A028982.

Cf. A084738, A065854, A128164.

Subsequence of A006532.

Sequence in context: A355626 A274567 A137994 * A116009 A068562 A123656

Adjacent sequences: A074383 A074384 A074385 * A074387 A074388 A074389

Keyword

nonn,bref,more

Author

Labos Elemer, Aug 22 2002

Extensions

Definition corrected by Juan Lopez, May 26 2008

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

Status

approved