

A006532


Numbers whose sum of divisors is a square.
(Formerly M3089)


54



1, 3, 22, 66, 70, 81, 94, 115, 119, 170, 210, 214, 217, 265, 282, 310, 322, 343, 345, 357, 364, 382, 385, 400, 472, 497, 510, 517, 527, 642, 651, 679, 710, 742, 745, 782, 795, 820, 862, 884, 889, 930, 935, 966, 970, 1004, 1029, 1066, 1080, 1092, 1146
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

If a and b are in the sequence and relatively prime, then a*b is also in the sequence.  Franklin T. AdamsWatters, Jan 12 2009
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
See the Beukers, Luca and Oort link for a proof that the sequence is infinite.  Robert Israel, Oct 15 2017


REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 8.
J.M. De Koninck, Ces nombres qui nous fascinent, Entry 94, p. 33, Ellipses, Paris 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Abraham Verghese, Cutting for Stone: A Novel. New York: Alfred A. Knopf, 2009, p.361, p. 528 largeprint edition.
David Wells, Curious and interesting numbers, Penguin Books, p. 111.


LINKS

Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373380.


FORMULA



EXAMPLE

3 is in the sequence because its divisors are 1 and 3, which add up to 4 = 2^2.
22 is in the sequence because its divisors are 1, 2, 11, 22, which add up to 36 = 6^2.
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.


MAPLE

for i from 1 to 1000 do if issqr(sigma(i)) then print(i); fi; od;


MATHEMATICA

Select[ Range[ 1150 ], IntegerQ[ Sqrt[ DivisorSigma[ 1, # ] ] ]& ]


PROG

(Haskell)
a006532 n = a006532_list !! (n1)
a006532_list = filter ((== 1) . a010052 . a000203) [1..]
(Magma) [n: n in [1..2000]  IsSquare(&+(Divisors(n)))]; // Vincenzo Librandi, May 31 2015


CROSSREFS



KEYWORD

nonn,nice


AUTHOR



EXTENSIONS



STATUS

approved



