Hi, I'm trying to compute the Juggler Sequence for relatively large numbers. Standard python doesn't calculate these correctly; I've discovered that Mathematica does, but I'm more familiar with Python and someone suggested I try out Sage.
I've tried this a couple ways using SageCell, and haven't yet succeeded. Here's the simplest possible implementation:
def juggle(n):
if n % 2:
return floor(sqrt(n)**3)
return floor(sqrt(n))
When I run this on 37, it works perfectly:
x = 37
print x,
while x > 1:
x = juggle(x)
print u"\u2192 {}".format(x),
produces
37 → 225 → 3375 → 196069 → 86818724 → 9317 → 899319 → 852846071 → 24906114455136 → 4990602 → 2233 → 105519 → 34276462 → 5854 → 76 → 8 → 2 → 1
which standard python fails (it finds 3374 instead of 3375). However, if I try a hard one, say, x = 48443, it times out after a few iterations:
48443 → 10662193 → 34815273349 → 6496130099313865 → 523578821252958052233532 → 723587466207 → 615512041010804067 → 482897358660562651148793788 → 21974925680433 → 103012783516625098121 → 1045530445028727953685811220915
I can improve this by using logarithms:
def juggle(n):
if n % 2:
return floor(exp(1.5*log(n)))
return floor(sqrt(n))
This implementation is much faster and gets much farther; however, it ends with an error:
TypeError: ECL says: #<a FLOATING-POINT-OVERFLOW>
I'd like to attach the full output as a text file, because it involves some very large numbers, but I don't have the karma yet. The number which causes the code to crash is about 3300 characters long. Here's the traceback, at least:
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-1-a9218cbdcf1b> in <module>()
8
9 while x > Integer(1):
---> 10 x = juggle(x)
11 print u"\u2192 {}".format(x),
<ipython-input-1-a9218cbdcf1b> in juggle(n)
1 def juggle(n):
2 if n % Integer(2):
----> 3 return floor(exp(RealNumber('1.5')*log(n)))
4 return floor(sqrt(n))
5
/home/sc_serv/sage/local/lib/python2.7/site-packages/sage/functions/other.pyc in __call__(self, x, **kwds)
412
413 try:
--> 414 return floor(SR(x).full_simplify().canonicalize_radical())
415 except ValueError:
416 pass
/home/sc_serv/sage/src/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.simplify_full (build/cythonized/sage/symbolic/expression.cpp:53709)()
9698 x = self
9699 x = x.simplify_factorial()
-> 9700 x = x.simplify_rectform()
9701 x = x.simplify_trig()
9702 x = x.simplify_rational()
/home/sc_serv/sage/src/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.simplify_rectform (build/cythonized/sage/symbolic/expression.cpp:55646)()
9846
9847 """
-> 9848 simplified_expr = self.rectform()
9849
9850 if complexity_measure is None:
/home/sc_serv/sage/src/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.rectform (build/cythonized/sage/symbolic/expression.cpp:53186)()
9530
9531 """
-> 9532 return self.maxima_methods().rectform()
9533
9534 def unhold(self, exclude=None):
/home/sc_serv/sage/local/lib/python2.7/site-packages/sage/symbolic/maxima_wrapper.pyc in __call__(self, *args, **kwds)
30 """
31 return super(MaximaFunctionElementWrapper, self).__call__(*args,
---> 32 **kwds).sage()
33
34 class MaximaWrapper(SageObject):
/home/sc_serv/sage/local/lib/python2.7/site-packages/sage/interfaces/interface.pyc in __call__(self, *args, **kwds)
655
656 def __call__(self, *args, **kwds):
--> 657 return self._obj.parent().function_call(self._name, [self._obj] + list(args), kwds)
658
659 def help(self):
/home/sc_serv/sage/local/lib/python2.7/site-packages/sage/interfaces/interface.pyc in function_call(self, function, args, kwds)
576 [s.name() for s in args],
577 ['%s=%s'%(key,value.name()) for key, value in kwds.items()])
--> 578 return self.new(s)
579
580 def _function_call_string(self, function, args, kwds):
/home/sc_serv/sage/local/lib/python2.7/site-packages/sage/interfaces/interface.pyc in new(self, code)
345
346 def new(self, code):
--> 347 return self(code)
348
349 ###################################################################
/home/sc_serv/sage/local/lib/python2.7/site-packages/sage/interfaces/interface.pyc in __call__(self, x, name)
280
281 if isinstance(x, string_types):
--> 282 return cls(self, x, name=name)
283 try:
284 return self._coerce_from_special_method(x)
/home/sc_serv/sage/local/lib/python2.7/site-packages/sage/interfaces/interface.pyc in __init__(self, parent, value, is_name, name)
695 self._name = parent._create(value, name=name)
696 except (TypeError, RuntimeError, ValueError) as x:
--> 697 raise TypeError(x)
698
699 def _latex_(self):
TypeError: ECL says: #<a FLOATING-POINT-OVERFLOW>
Does anyone have any suggestions for me? Thanks in advance!
