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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.
3
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
Name and offset corrected by Amiram Eldar, Aug 15 2019
STATUS
approved