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A008848 Squares whose sum of divisors is a square. 13
1, 81, 400, 32400, 1705636, 3648100, 138156516, 295496100, 1055340196, 1476326929, 2263475776, 2323432804, 2592846400, 2661528100, 7036525456, 10994571025, 17604513124, 39415749156, 61436066769, 85482555876, 90526367376, 97577515876, 98551417041 (list; graph; refs; listen; history; text; internal format)
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
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) 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
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
EXAMPLE
n=81: sigma(81) = 1+3+9+27+81 = 121 = 11^2.
n=32400: 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
a(n) = A008847(n)^2.
Subsequence of A000290, of A006532, and of A069070.
Sequence in context: A017498 A097025 A074387 * A237182 A237176 A357015
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 19 03:21 EDT 2024. Contains 370952 sequences. (Running on oeis4.)