A173792 - OEIS

%I #11 Jul 09 2012 12:20:50

%S 9,14,19,51,99,243,339,579,723,1059,1640,1683,1923,2739,3363,3699,

%T 4419,5619,6963,7443,8979,10083,10659,12483,13779,15843,18819,20403,

%U 21219,22899,23763,25539,32259,34323,37539,38643,44403,45603,49299,53139,55779

%N Numbers of the form x^2 + y^2 + z^2 = phi(x*y*z) + sigma(x*y*z).

%C Phi = A000010 is Euler's totient and sigma = A000203 is the sum of divisors.

%C Let p prime, then (x,y,z) = (1,p,p),(p,1,p),(p,p,1) are solutions because phi(p^2) + sigma(p^2) = (p-1)p + p(p+1)+1 = 2p^2 + 1.

%H Donovan Johnson, <a href="/A173792/b173792.txt">Table of n, a(n) for n = 1..1000</a>

%e 9 is in the sequence because for (x,y,z) = (1,2,2), x^2 + y^2 + z^2 = 9, phi(4)=2, sigma(4)=7, and phi(4) + sigma(4) = 9 ;

%e 1640 is in the sequence because for (x,y,z) = (6,2,40), x^2 + y^2 + z^2 = 1640, phi(480)=128, sigma(480)=1512, and phi(480) + sigma(480) = 1640.

%p isA173792 := proc(n)

%p         for x from 1 do

%p                 if x^2 > n then

%p                         return false;

%p                 end if;

%p                 for y from x do

%p                         if x^2+y^2 > n then

%p                                 break;

%p                         end if;

%p                         if issqr(n-x^2-y^2) then

%p                                 z := sqrt(n-x^2-y^2) ;

%p                                 p := x*y*z ;

%p                                 if n = numtheory[sigma](p) + numtheory[phi](p) then

%p                                         return true;

%p                                 end if;

%p                         end if;

%p                 end do:

%p         end do:

%p end proc:

%p for n from 1 do

%p         if isA173792(n) then

%p                 printf("%d,\n",n) ;

%p         end if;

%p end do: # _R. J. Mathar_, Jul 08 2012

%K nonn

%O 1,1

%A _Michel Lagneau_, Feb 24 2010