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A185352 The "smallest countdown" numbers are the smallest positive integer that cannot be made using the numbers n through 1, in order, using the operations +, -, *, /, and parentheses.

%I A185352 #38 Jun 08 2024 15:44:07
%S A185352 2,4,8,17,39,92,275,922,2894,10843,35944
%N A185352 The "smallest countdown" numbers are the smallest positive integer that cannot be made using the numbers n through 1, in order, using the operations +, -, *, /, and parentheses.
%C A185352 Inspired by a now-lost blog post in which someone discussed a "new year's countdown" equation for 2012, e.g., 10 * (9 + ((8 * (((7 + (6 / (5 * 4))) * 3) + 2)) + 1)) = 2012. This sequence has been "verified" by two independently created programs.
%e A185352 for n = 3, a(3) = 8, because 3*2+1=7, and 3*(2+1)=9, but there is no equation with 3,2,and 1 in order that equals 8. Note that if we allow the order to change, we can make 8, because 2*(3+1)=8, but reordering is not allowed.
%o A185352 (Python)
%o A185352 from fractions import Fraction
%o A185352 def genAllTrees(l):
%o A185352     if len(l) == 0:
%o A185352         return
%o A185352     elif len(l) == 1:
%o A185352         yield l[0], str(l[0])
%o A185352     else:
%o A185352         for middle in range(len(l)):
%o A185352             for lval, leqn in genAllTrees(l[:middle]):
%o A185352                 for rval, reqn in genAllTrees(l[middle:]):
%o A185352                     yield lval+rval, ("(" + leqn + " + " + reqn + ")")
%o A185352                     yield lval-rval, ("(" + leqn + " - " + reqn + ")")
%o A185352                     yield lval*rval, ("(" + leqn + " * " + reqn + ")")
%o A185352                     if rval != Fraction(0):
%o A185352                         yield lval/rval, ("(" + leqn + " / " + reqn + ")")
%o A185352 def findSmallestIntNotPresent(n):
%o A185352     vals = {}
%o A185352     for val, eqn in genAllTrees([Fraction(i) for i in range(n, 0, -1)]):
%o A185352         if val.denominator == 1:
%o A185352             val = val.numerator
%o A185352             if val not in vals:
%o A185352                 vals[val] = eqn
%o A185352     i = 1
%o A185352     while i in vals:
%o A185352         i += 1
%o A185352     return i
%o A185352 for i in range(1, 11):
%o A185352     print(i, findSmallestIntNotPresent(i))
%Y A185352 Related to A060315, which is the smallest number that cannot be made with the numbers 1 to n, in any order.
%K A185352 nonn,hard,more
%O A185352 1,1
%A A185352 _Peter Boothe_ and Abraham Asfaw, Feb 08 2012
%E A185352 a(10)-a(11) from _Hiroaki Yamanouchi_, Oct 04 2014