The values $n(p)$ for primes $13 \leq p < 100$, found by computation: $n(13) = n(17) = 716$, $n(19) = 62987$, $n(23) = 367082$, $n(29) = 728366$, $n(31) = 64822396$, $n(37) = 1306238012$, $n(41) = 11182598506$, $n(43) = 715041747422$, $n(47) = 51913478860882$, $n(53) = 454746157008782$, $n(59) = 9314160363311806$, $n(61) = n(67) = 261062105979210901$, $n(71) = 696537082207206753592$, $n(73) = 54097844397380813592487$, $n(79) = 286495021083846822067822$, $n(83) = 80126789479717708423427656$, $n(89) = 1560127578864999430859224576$, $n(97) = 161426380685234430031618378951$.

The values $n(p)$ for primes $13 \leq p < 100$, found by computation: $n(13) = n(17) = 716$, $n(19) = 62987$, $n(23) = 367082$, $n(29) = 728366$, $n(31) = 64822396$, $n(37) = 1306238012$, $n(41) = 11182598506$, $n(43) = 715041747422$, $n(47) = 51913478860882$, $n(53) = 454746157008782$, $n(59) = 9314160363311806$, $n(61) = n(67) = 261062105979210901$, $n(71) = 696537082207206753592$, $n(73) = 54097844397380813592487$, $n(79) = 286495021083846822067822$, $n(83) = 80126789479717708423427656$, $n(89) = 1560127578864999430859224576$, $n(97) = 161426380685234430031618378951$.

Edit: For completeness -- the GAP code for computing these values is as follows:

n := function ( x )

  local  primes, remainders, solutions;

  primes     := Filtered([3..x],IsPrime);
  remainders := Tuples([1,2],Length(primes));
  solutions  := List(remainders,rem->ChineseRem(primes,rem));
  return Minimum(Difference(solutions,[1,2]));
end;

Just some values found by computation: $n(13) = n(17) = 716$,  $n(19) = 62987$, $n(23) = 367082$, $n(29) = 728366$, $n(31) = 64822396$, $n(37) = 1306238012$, $n(41) = 11182598506$, $n(43) = 715041747422$, $n(47) = 51913478860882$, $n(53) = 454746157008782$, $n(59) = 9314160363311806$, $n(61) = n(67) = 261062105979210901$, $n(71) = 696537082207206753592$ and $n(73) = 54097844397380813592487$.

The values $n(p)$ for primes $13 \leq p < 100$, found by computation: $n(13) = n(17) = 716$,  $n(19) = 62987$, $n(23) = 367082$, $n(29) = 728366$, $n(31) = 64822396$, $n(37) = 1306238012$, $n(41) = 11182598506$, $n(43) = 715041747422$, $n(47) = 51913478860882$, $n(53) = 454746157008782$, $n(59) = 9314160363311806$, $n(61) = n(67) = 261062105979210901$, $n(71) = 696537082207206753592$, $n(73) = 54097844397380813592487$, $n(79) = 286495021083846822067822$, $n(83) = 80126789479717708423427656$, $n(89) = 1560127578864999430859224576$, $n(97) = 161426380685234430031618378951$.