n through n+7 have same d() / A049053

[Hugo Pfoertner]

Jud McCranie wrotte:
> 
> At 04:16 AM 11/2/2004, Ralf Stephan wrote:
> >Hello,
> >A049053 gives all numbers n where n,n+1,...,n+6 have the same number
> >of divisors. Allan Swett gave a counterexample to my conjecture that
> >n is 5 mod 16 while searching for solutions to the problem with n...n+7.
> >
> >He asks:
> > > I was hoping to find 8 consecutive having same d()... I
> > > didn't spot evidence of this so far... do you know if this is impossible?
> 
> It is not impossible, but it gets a lot harder for 8 in a row because you
> have one number that is divisible by 8 and one that is divisible by 4 but
> not divisible by 8.  There are solutions with 9 in a row, I'm pretty sure
> they are in Unsolved Problems in Number Theory.

Yes it is there: B18 Solutions of d(n)=d(n+1), page 111 in 3rd edition.

An example is given for a sequence of 9 consecutive numbers with 48
divisors, discovered in 1990 by Ivo Duentsch and Roger Eggleton:

17796126877482329126044 = 2 ^ 2 x 7 x 4327 x 456293 x 321911699243
17796126877482329126045 = 5 x 17 ^ 2 x 47 x 53 x 4944062119125691
17796126877482329126046 = 2 x 3 ^ 2 x 179 x 5171 x 1068133213285183
17796126877482329126047 = 11 ^ 5 x 23 x 107 x 44900425217777
17796126877482329126048 = 2 ^ 5 x 19 x 4590338339 x 6376424429
17796126877482329126049 = 3 x 13 ^ 2 x 241 x 557 x 261484106225711
17796126877482329126050 = 2 x 5 ^ 2 x 11831 x 189043 x 159137830837
17796126877482329126051 = 7 ^ 5 x 29 x 351121 x 103987345177
17796126877482329126052 = 2 ^ 2 x 3 x 149 x 991723 x 10036160394373

It is unknown if this is the smallest example.

And of course:
http://www.research.att.com/projects/OEIS?Anum=A006558

Hugo Pfoertner