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A074386
Numbers n such that sigma(n) is the square of a prime.
2
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
M. A. Bennett and A. Levin, The Nagell-Ljunggren equation via Runge’s method, Monatshefte für Mathematik 177:1 (2015), 15-31.
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
Subsequence of A006532.
Sequence in context: A355626 A274567 A137994 * A116009 A068562 A123656
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