OFFSET
1,2
COMMENTS
Solutions to sigma(x^2) = (2k+1)^2. - Labos Elemer, Aug 22 2002
Intersection of A006532 and A000290. The product of any two coprime terms is also in this sequence. - Charles R Greathouse IV, May 10 2011
Conjectures: (1) a(2) = 81 is the only prime power (A246655) in this sequence. (2) 81 and 400 are only terms x for which sigma(x) is in A246655. (3) x = 1 is the only such term that sigma(x) is also a term. See also comments in A074386, A336547 and A350072. - Antti Karttunen, Jul 03 2023, (2) corrected in May 11 2024
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 10.
I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..400
FORMULA
a(n) = A008847(n)^2.
EXAMPLE
n=81: sigma(81) = 1+3+9+27+81 = 121 = 11^2.
n=400: sigma(400) = sigma(16)*sigma(25) = 31*31 = 961.
n=32400 (= 81*400): sigma(32400) = 116281 = 341^2 = 121*961.
MATHEMATICA
Do[s=DivisorSigma[1, n^2]; If[IntegerQ[Sqrt[s]]&&Mod[s, 2]==1, Print[n^2]], {n, 1, 10000000}] (* Labos Elemer *)
Select[Range[320000]^2, IntegerQ[Sqrt[DivisorSigma[1, #]]]&] (* Harvey P. Dale, Feb 22 2015 *)
PROG
(PARI) for(n=1, 1e6, if(issquare(sigma(n^2)), print1(n^2", "))) \\ Charles R Greathouse IV, May 10 2011
CROSSREFS
Terms of A008847 squared.
KEYWORD
nonn
AUTHOR
STATUS
approved