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A008848
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Squares whose sum of divisors is a square.
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13
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1, 81, 400, 32400, 1705636, 3648100, 138156516, 295496100, 1055340196, 1476326929, 2263475776, 2323432804, 2592846400, 2661528100, 7036525456, 10994571025, 17604513124, 39415749156, 61436066769, 85482555876, 90526367376, 97577515876, 98551417041
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OFFSET
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1,2
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COMMENTS
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Solutions to sigma(x^2) = (2k+1)^2. - Labos Elemer, Aug 22 2002
Conjectures: (1) a(2) = 81 is the only prime power (A246655) in this sequence. (2) It is also the only term whose square is a prime power. (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
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REFERENCES
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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.
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LINKS
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EXAMPLE
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n=81: sigma(81) = 1+3+9+27+81 = 121 = 11^2.
n=32400: sigma(32400) = 116281 = 341^2 = 121*961.
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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