Why 82,000 is an extraordinary number - Numberphile
Has anyone here watched the Numberphile video about 82000 or heard about the sequence (2,3,4,82000)? Each member, a(n), is the smallest integer greater than 1 that can be written with only 1s and 0s in bases 2 through n. No one has found a(6) and it's even conjectured not to exist. Apparently the search has gone up to around 10^10^7.
Naturally, my first instinct was that this problem looks very similar to other problems in googology. Has anyone else looked into this? If someday this sequence is proved not to end then I think we might have something decent on our hands! And if it really is total, I would be very curious about how quickly it grows.
What do you guys think? Also, I've written an obfuscated algorithm to generate this sequence not very surprising coming from me for anyone who's interested.
def a(n):
def p(x,b):
d=1
while x:
d*=x%b<2
x//=b
return d
N=2
while 0 in [p(N,i) for i in range(2,n+1)]:N+=1
return N
Aaaand just for the heck of it, here's some slightly more readable and much more efficient code.
def powerSum(n):
def binExp(x,b):
s=bin(x)[-1:1:-1]
return sum([int(s[i])*b**i for i in range(len(s))])
def isPowerSum(num,base):
rem=True
while rem and num:
rem*= num%base<2
num//= base
return rem
N=2
valid=False
while not valid:
b=n-1
valid=True
while valid and 2<b:
valid*=isPowerSum(binExp(N,n),b)
b-=1
if not valid:N+=1
return binExp(N,n)