restart:with(numtheory):

# formula for efficiency
eff := (p,a) -> evalf[30](ln(sigma(p^a)/p^a)/ln(p^a)):

# forbidden factors
x := 4*127*19*7*11*331*97*61*31*13*398581*1093*3*5*307*17:

# finding the smallest exponent "a" such that gcd(sigma(p^a), x) = 1
n := 0:
for p from 3 to 10000 by 2 do if gcd(p,x) = 1 and isprime(p) then 
 a := 1;
 while gcd(sigma(p^a), x) > 2 do a := a+1; od;
 n := n+1;
 t[n] := (p,a,eff(p,a));
fi;
od;

# sorting by efficiency
for i from 1 to n do 
 im := i;
 for j from i+1 to n do if t[j][3] > t[im][3] then im := j; fi; od;
 m := t[im]; t[im] := t[i]; t[i] := m;
od:

# computing the product of components with highest efficiency while abundancy < 2
abun := 1:
prod := 1:
for i from 1 while abun*(1+1/(t[i][1]-1)) < 2 do 
 abun := abun*(1+1/(t[i][1]-1));
 prod := prod*t[i][1]^t[i][2];
od:
print(evalf(abun), evalf(log[10](prod)));

### result: 1.997859878, 1735.673672