In 2002 Jack Brennen asked in
http://groups.yahoo.com/group/primenumbers/message/9879 :
> For a given n>=2, what is the largest n-tuple of
> consecutive integers for which the prime factorization
> of each member is known?

This is now answered at the new record page
"Largest Consecutive Factorizations" at
http://hjem.get2net.dk/jka/math/consecutive_factorizations.htm
Only cases with at least 500 digits are listed. This currently means n<=7.
(I call it k instead of n).

Jack also wrote:
> So I'm curious, what is the longest titanic factorized n-tuple?
> The above-mentioned 5-tuple is titanic. Can someone give me
> 6 consecutive titanic numbers along with their prime factorizations?

It took 5 years but then he gets one more than requested:
m to m+6 for m = 21247003564*2411#-1 with 1037 digits.
Prime factorizations where pN is an N-digit prime:

m = 12906420959*p1027
m+1 = 2^2*103*51570397*2411#, where 2411# = 2*3*5*7*...*2411
m+2 = 2524541*p1031
m+3 = 2*5002841*6245491249*p1020
m+4 = 3*485475518243*p1025
m+5 = 2^2*(5311750891*2411#+1)
m+6 = 5*3691*23063^2*2961991*19076087*3778442561*p1001

PrimeForm/GW made prp tests and Marcel Martin's Primo proved
p1027, p1031, p1020, p1025, p1001.

(m+5)/4 = 5311750891*2411#+1 = p1037 was found by Markus Frind
and Paul Underwood with NewPGen and PrimeForm/GW in 2003
during an AP8 search:
http://tech.groups.yahoo.com/group/primenumbers/message/12734
They found more than 10 million prp's and kindly gave me access
to 7278744 stored ones so I could search for 7 consecutive titanic
factorizations with limited computing power.

Each of the prp's are of form x+1 = k*2411#+1 for small k, so x
is trivially factored. If x and x+1 are both factored then so are nx
and nx+n for small n, so (x,x+1) can potentially be used for
7 consecutive factorizations in 21 ways:
6 ways around (x,x+1), 5 ways around (2x,2x+2) if 2x+1 is also
factored, 4 ways around (3x,3x+3), 3 around (4x,4x+4),
2 around (5x,5x+5), and 1 last way from 6x to 6x+6.

The x values are part of the same arithmetic progression k*2411#.
This made it possible to sieve all possibilities to 10^12 in the same
sieve run with my APTreeSieve. It stored found factors and after
dividing by them, the cofactors of x-1 and 2x+1 were prp tested.
If one of them is prp then there are 3 consecutive factorizations
(x-1,x,x+1) or (2x,2x+1,2x+2), giving 9 possibilities of extending to 7.
If x-1 and 2x+1 remained unfactored then that x was skipped in
order to spare further cofactor prp tests which would give much
smaller chance of extending to 7 numbers.

The single solution with 7 numbers is 4x-1 to 4x+5 for
x = 5311750891*2411#.
There were many with 6 numbers out of 7, and lots with 5.
Limited GMP-ECM found many factors above 10^12 but no
complete factorizations to reach 7.
k=8 would be reached if any of these 5 composites were complete factored:

(21247003564*2411#-2)/(2*63501029*1676569836074094735760183)
(21247003564*2411#+6)/(2*3^3*44041*50036011*18714589754509974127923506831749)
(9902172078*2411#+3)/3
(10520194890*2411#+3)/(3*137809990746376231)
(9988354978*2411#+3)/(3*31850135205297880082057*623786556972123391792124611*1426\
15599612981064331297)

All 5 have had the recommended GMP-ECM work for 30-digit factors:
453 curves with B1=250000.
The first 2 are at the ends of the k=7 record. The last 3 are holes
in cases with 7 of 8 factored. All 3 happen to be of form 2x+3.

The search for k=7 was originally planned for June 2005 together
with Décio Luiz Gazzoni Filho after discussion at
http://tech.groups.yahoo.com/group/primeform/message/5975?l=1
In fact, this post is a delayed follow-up to that post.
Décio did not have time and I did it alone after getting a faster
computer this year.

The records for k=2 and k=3 are the Mersenne and twin prime records.
The records for k=4 and k=5 were found by starting with known
Cunningham chains in The Prime Pages database and factoring a
single additional number.
If (p, 2p+1, 4p+3) is a CC3 with p+1 factored by construction,
then (4p, 4p+2, 4p+3, 4p+4) is factored, and only 4p+1 is missing for k=5.
For k=4 there is both a 2063-digit record with proven prime factors
(the same as k=5), and a 4187-digit record with a prp factor
(240819405*2^13879+3)/(3*13*43*358877).
A certification of that would give shared record credit.
The record for k=6 is the same as k=7.

The record page is mentioned at
http://mersenneforum.org/showthread.php?p=111804#post111804
where some people object to prime numbers being allowed as
their own prime factorization.

--
Jens Kruse Andersen