A006532 Numbers whose sum of divisors is a square.

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

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• nice

Comments

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

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

Examples

			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.
		

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 large-print edition.
• David Wells, Curious and interesting numbers, Penguin Books, p. 111.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.
• J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
• Index entries for sequences related to sums of divisors

Crossrefs

Cf. A074385, A000203, A020477.

Programs

Haskell

a006532 n = a006532_list !! (n-1)
a006532_list = filter ((== 1) . a010052 . a000203) [1..]
-- Reinhard Zumkeller, Jun 09 2013

Magma

[n: n in [1..2000] | IsSquare(&+(Divisors(n)))]; // Vincenzo Librandi, May 31 2015

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, # ] ] ]& ]

PARI

is(n)=issquare(sigma(n)) \\ Charles R Greathouse IV, Jun 05 2013

Sage

[n for n in (1..1000) if sigma(n).is_square()] # Giuseppe Coppoletta, Dec 16 2014

Formula

A010052(A000203(a(n))) = 1. - Reinhard Zumkeller, Jun 09 2013

Extensions

a(42)-a(51) from Enoch Haga, circa 1999