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A173792
Numbers of the form x^2 + y^2 + z^2 = phi(x*y*z) + sigma(x*y*z).
1
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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 24 2010
STATUS
approved