A173792 Numbers of the form x^2 + y^2 + z^2 = phi(x*y*z) + sigma(x*y*z).

9, 14, 19, 51, 99, 243, 339, 579, 723, 1059, 1640, 1683, 1923, 2739, 3363, 3699, 4419, 5619, 6963, 7443, 8979, 10083, 10659, 12483, 13779, 15843, 18819, 20403, 21219, 22899, 23763, 25539, 32259, 34323, 37539, 38643, 44403, 45603, 49299, 53139, 55779

Offset

1,1

Comments

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

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.

Links

Donovan Johnson, Table of n, a(n) for n = 1..1000

Example

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 ;
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.

Maple

isA173792 := proc(n)
        for x from 1 do
                if x^2 > n then
                        return false;
                end if;
                for y from x do
                        if x^2+y^2 > n then
                                break;
                        end if;
                        if issqr(n-x^2-y^2) then
                                z := sqrt(n-x^2-y^2) ;
                                p := x*y*z ;
                                if n = numtheory[sigma](p) + numtheory[phi](p) then
                                        return true;
                                end if;
                        end if;
                end do:
        end do:
end proc:
for n from 1 do
        if isA173792(n) then
                printf("%d, \n", n) ;
        end if;
end do: # R. J. Mathar, Jul 08 2012

Crossrefs

Sequence in context: A171123 A327896 A302056 * A332588 A034703 A006624

Adjacent sequences: A173789 A173790 A173791 * A173793 A173794 A173795

Keyword

nonn

Author

Michel Lagneau, Feb 24 2010

Status

approved