Problems & Puzzles: Puzzles

Puzzle 886. The earliest largest set  of consecutive integers type k-almost primes

A natural number is called k-almost prime if it has exactly k prime factors, counted with multiplicity.

The smallest integer type k-almost prime is 2^k

A set of consecutive integers type k-almost prime with the largest quantity of members Lmax is a set with Lmax=2^k-1 members.

Here are some of these known sets:

k Lmax Start of the earliest largest set of Lmax consecutive integers type k-almost prime
2 3 33
3 7 211673
4 15 97524222465
5 31 ?

 Which means that:

k=2, Lmax=3
33 34 35
3*11 2*17 5*7

 

k=3, Lmax=7 (See A067813)
211673 211674 211675 211676 211677 211678 211679
7*11*2749 2*3*35279 5^2*8467 2^2*52919 3*37*1907 2*109*971 13*19*857

 

k=4, L=15 (See A067814)
97524222465 97524222466 97524222467 97524222468 97524222469 97524222470 97524222471
3*5*42751*152081 2*11*19*233311537 7*29*149*3224261 2^2*3*8127018539 73^2*251*72911 2*5*67*145558541 3^2*13*833540363
             
97524222472 97524222473 97524222474 97524222475 97524222476 97524222477 97524222478
2^3*12190527809 17^2*3499*96443 2*3*7*2322005297 5^2*18493*210943 2^2*23029*1058711 3*11*18457*160117 2*23*151*14040343
             
97524222479            
47*181^2*63337            

 

For k=5, Lmax=31 but the best already  known is L=15 and the start integer is 24819420480104 (by Brian Trial, May 2017. See A067820)

Q1. Can you add some terms to the sequence A067820?

Q2. Can you produce a set of 31 consecutive integers 5-almost primes, no matter if your set is not the earliest one?

 


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