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
Keywords
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)
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
Comments