A008848 Squares whose sum of divisors is a square.

1, 81, 400, 32400, 1705636, 3648100, 138156516, 295496100, 1055340196, 1476326929, 2263475776, 2323432804, 2592846400, 2661528100, 7036525456, 10994571025, 17604513124, 39415749156, 61436066769, 85482555876, 90526367376, 97577515876, 98551417041

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

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

Also intersection of A069070 and A000290. - Michel Marcus, Oct 06 2013

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

Examples

			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.
		

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

Crossrefs

Terms of A008847 squared.

Cf. A028982, A001248, A000203, A074386, A231484 (odd terms), A246655, A336547, A350072.

Subsequence of A000290, of A006532, and of A069070.

Programs

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 *)

PARI

for(n=1,1e6,if(issquare(sigma(n^2)), print1(n^2", "))) \\ Charles R Greathouse IV, May 10 2011

Formula

a(n) = A008847(n)^2.