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%I #38 May 11 2024 16:27:29
%S 1,81,400,32400,1705636,3648100,138156516,295496100,1055340196,
%T 1476326929,2263475776,2323432804,2592846400,2661528100,7036525456,
%U 10994571025,17604513124,39415749156,61436066769,85482555876,90526367376,97577515876,98551417041
%N Squares whose sum of divisors is a square.
%C Solutions to sigma(x^2) = (2k+1)^2. - _Labos Elemer_, Aug 22 2002
%C Intersection of A006532 and A000290. The product of any two coprime terms is also in this sequence. - _Charles R Greathouse IV_, May 10 2011
%C Also intersection of A069070 and A000290. - _Michel Marcus_, Oct 06 2013
%C 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 A336547 and A350072. - _Antti Karttunen_, Jul 03 2023, (2) corrected in May 11 2024
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 10.
%D I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.
%H Donovan Johnson, <a href="/A008848/b008848.txt">Table of n, a(n) for n = 1..400</a>
%e n=81: sigma(81) = 1+3+9+27+81 = 121 = 11^2.
%e n=400: sigma(400) = sigma(16)*sigma(25) = 31*31 = 961.
%e n=32400 (= 81*400): sigma(32400) = 116281 = 341^2 = 121*961.
%t Do[s=DivisorSigma[1, n^2]; If[IntegerQ[Sqrt[s]]&&Mod[s, 2]==1, Print[n^2]], {n, 1, 10000000}] (* _Labos Elemer_ *)
%t Select[Range[320000]^2,IntegerQ[Sqrt[DivisorSigma[1,#]]]&] (* _Harvey P. Dale_, Feb 22 2015 *)
%o (PARI) for(n=1,1e6,if(issquare(sigma(n^2)), print1(n^2", "))) \\ _Charles R Greathouse IV_, May 10 2011
%Y a(n) = A008847(n)^2.
%Y Cf. A028982, A001248, A000203, A231484 (odd terms), A246655, A336547, A350072.
%Y Subsequence of A000290, of A006532, and of A069070.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
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