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%I M3089 #80 Sep 08 2022 08:44:35
%S 1,3,22,66,70,81,94,115,119,170,210,214,217,265,282,310,322,343,345,
%T 357,364,382,385,400,472,497,510,517,527,642,651,679,710,742,745,782,
%U 795,820,862,884,889,930,935,966,970,1004,1029,1066,1080,1092,1146
%N Numbers whose sum of divisors is a square.
%C If a and b are in the sequence and relatively prime, then a*b is also in the sequence. - _Franklin T. Adams-Watters_, Jan 12 2009
%C Apart from a(2), all terms are composite. Bunyakovsky's conjecture implies that this sequence is infinite, since then (e.g.) there are infinitely many primes of the form p = 3k^2 - 1, whence sigma(2p) = 3p + 3 = 9k^2. - _Charles R Greathouse IV_, May 12 2011
%C See the Beukers, Luca and Oort link for a proof that the sequence is infinite. - _Robert Israel_, Oct 15 2017
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 8.
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 94, p. 33, Ellipses, Paris 2008.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D Abraham Verghese, Cutting for Stone: A Novel. New York: Alfred A. Knopf, 2009, p.361, p. 528 large-print edition.
%D David Wells, Curious and interesting numbers, Penguin Books, p. 111.
%H T. D. Noe, <a href="/A006532/b006532.txt">Table of n, a(n) for n = 1..10000</a>
%H Frits Beukers, Florian Luca and Frans Oort, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.119.05.373">Power Values of Divisor Sums</a>, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.
%H J. Meeus & N. J. A. Sloane, <a href="/A006532/a006532_1.pdf">Correspondence, 1974-1975</a>
%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A010052(A000203(a(n))) = 1. - _Reinhard Zumkeller_, Jun 09 2013
%e 3 is in the sequence because its divisors are 1 and 3, which add up to 4 = 2^2.
%e 22 is in the sequence because its divisors are 1, 2, 11, 22, which add up to 36 = 6^2.
%e 32 is not in the sequence, because its divisors, 1, 2, 4, 8, 16, 32, add up to 63, which is one short of 8^2.
%p for i from 1 to 1000 do if issqr(sigma(i)) then print(i); fi; od;
%t Select[ Range[ 1150 ], IntegerQ[ Sqrt[ DivisorSigma[ 1, # ] ] ]& ]
%o (PARI) is(n)=issquare(sigma(n)) \\ _Charles R Greathouse IV_, Jun 05 2013
%o (Haskell)
%o a006532 n = a006532_list !! (n-1)
%o a006532_list = filter ((== 1) . a010052 . a000203) [1..]
%o -- _Reinhard Zumkeller_, Jun 09 2013
%o (Sage) [n for n in (1..1000) if sigma(n).is_square()] # _Giuseppe Coppoletta_, Dec 16 2014
%o (Magma) [n: n in [1..2000] | IsSquare(&+(Divisors(n)))]; // _Vincenzo Librandi_, May 31 2015
%Y Cf. A074385, A000203, A020477.
%K nonn,nice
%O 1,2
%A _N. J. A. Sloane_, _Simon Plouffe_
%E a(42)-a(51) from _Enoch Haga_, circa 1999
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