A289385 - OEIS

%I #28 Mar 02 2019 04:22:09

%S 1,4,6,22,48,64,390,429,638,682,880,2013,2310,2320,2968,2970,3625,

%T 4692,5696,5865,7772,9213,9715,9944,10778,11254,12963,13039,13704,

%U 15180,23370,27236,27556,34045,34445,36141,44336,53136,54376,58548,73185,78952,105464

%N Nonprime numbers k such that the sum of the divisors of k^2 is of the form m^2 + m + 1.

%C The corresponding values of m are given by the sequence {0, 5, 9, 30, 81, 90, 718, 562, 900, 961, 1451, ...} containing the subsequence of perfect squares {0, 9, 81, 900, 961, 4624, 8100, ...}.

%C If k is prime, the sum of the divisors of k^2 is the trivial value k^2 + k + 1.

%H Giovanni Resta, <a href="/A289385/b289385.txt">Table of n, a(n) for n = 1..500</a> (first 127 terms from Robert G. Wilson v)

%e 22 is in the sequence because sigma(22^2) = 931 = 30^2 + 30 + 1.

%p with(numtheory):nn:=10^5:

%p for n from 1 to nn do:

%p    if type(n,prime)=false

%p     then

%p     y:=sqrt(4*sigma(n^2)-3):

%p      z :=(-1+y)/2:

%p      if y=floor(y) and z=floor(z) then printf(`%d, `, n):

%p        else

%p       fi:fi:

%p od:

%t fQ[n_] := ! PrimeQ@ n && Block[{sd = DivisorSigma[1, n^2]}, y = Sqrt[4sd -3]; z = (y -1)/2; y == Floor@ y && z == Floor@ z]; Select[Range@ 125491, fQ] (* _Robert G. Wilson v_, Jul 05 2017 *)

%Y Cf. A000203, A002061.

%K nonn

%O 1,2

%A _Michel Lagneau_, Jul 04 2017