A049226 Composite numbers n such that the sum of divisors of n, sigma(n), divided by the number of divisors, d(n) and sigma(n) minus n are both rational squares.

119, 527, 1196, 3591, 5831, 6887, 12319, 15407, 18575, 33271, 47959, 51119, 56853, 63119, 65151, 116399, 176911, 328151, 373319, 437999, 438311, 520319, 568519, 724687, 734111, 851927, 957551, 1059191, 1140071, 1437599, 1760831, 1813511, 2320919, 3354479, 3383420

Offset

1,1

Comments

The prime numbers with this property are primes of the form 2*k^2 - 1 (A066436). - Amiram Eldar, Aug 15 2019

The first terms for which the ratio sigma(n)/d(n) is not an integer are 267910912, 1398459816, and 1703794876. - Giovanni Resta, Aug 30 2019

Links

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 300 terms from Amiram Eldar)

Example

a(27) = 957551 is a term since the sum of its 16 divisors is sigma(957551) = 1166400 and both 1166400/16 = 72900 = 270^2 and 1166400 - 957551 = 208849 = 457^2 are perfect squares.

Mathematica

Select[Range[10^5], CompositeQ[#] && And @@ IntegerQ /@ Sqrt[{(s = DivisorSigma[1, #]) * DivisorSigma[0, #], s - #}] &] (* Amiram Eldar, Aug 15 2019 *)

Prog

(PARI) is(n) = my(f = factor(n), s = sigma(f), nd = numdiv(f)); issquare(s/nd) && issquare(s - n) && !isprime(n) \\ David A. Corneth, Aug 15 2019
(Magma) [m:m in [1..3400000]|not IsPrime(m) and IsSquare(SumOfDivisors(m)/#Divisors(m)) and IsSquare(SumOfDivisors(m)-m)]; // Marius A. Burtea, Aug 15 2019

Crossrefs

Cf. A000005, A000203, A049207, A049227, A049228, A048699, A066436.

Sequence in context: A103852 A157040 A256907 * A106572 A327655 A326795

Adjacent sequences: A049223 A049224 A049225 * A049227 A049228 A049229

Keyword

nonn

Author

Enoch Haga

Extensions

Name and offset corrected by Amiram Eldar, Aug 15 2019

Status

approved