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

Author

Enoch Haga

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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

Mathematica

Select[Range[10^5], CompositeQ[#] && And @@ IntegerQ /@ Sqrt[{(s = DivisorSigma[1, #]) * DivisorSigma[0, #], s - #}] &] (* Amiram Eldar, Aug 15 2019 *)
cnQ[n_]:=With[{sg=DivisorSigma[1,n]},CompositeQ[n]&&AllTrue[{Sqrt[sg/DivisorSigma[0,n]],Sqrt[sg-n]},IntegerQ]]; Select[Range[ 339*10^4],cnQ] (* Harvey P. Dale, Mar 31 2025 *)

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

Extensions

Name and offset corrected by Amiram Eldar, Aug 15 2019