Is A073071 correct?

[Peter Pein]

Dear seqfans,

propably I overlook something important, but I can't figure out what.

The definition of http://www.research.att.com/~njas/sequences/A073071 is:
"Least k such that k! > prime(1)*prime(2)*...*prime(n) where prime(n) is the
n-th prime."
and it starts with n=1.

let pp=prod(i=1,n,prime(i))

pp(1)=  2 and 2! <=  2 < 3!
pp(2)=  6 and 3! <=  6 < 4!
pp(3)= 30 and 4! <  30 < 5!
...

so A073071 should start 3,4,5... but it reads 2,4,6...

Using Mathematica, I get
3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28,
29, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 53,
54, 55, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77,
79, 80, 81, 83, 84, 85, 86, 88, 89, 90, 92, 93, 94, 96, 97, 98, 99, 101, 102,
103, 105, 106, 107, 108, 110, 111, 112, 114, 115, 116, 118, 119, 120, 121,
123, 124, 125, 127, 128, 129, 130
as the first 100 numbers.

To be sure, I verified this with pari/gp:

 pp=1;k=1;kf=1;forprime(p=2,prime(100),
   pp*=p;while(kf<=pp,k+=1;kf*=k);print1(k,","))
3,4,5,6,7,8,10,...,125,127,128,129,130

Where did I go wrong?

Peter