A358075 a(1) = 1; a(n+1) is the smallest integer > 0 that cannot be obtained from the integers {a(1), ..., a(n)} using each number exactly once and the operators +, -, *, /, where intermediate subexpressions must be integers.
1, 2, 4, 11, 34, 152, 1007, 6703, 56837, 766478
Offset: 1
Examples
All positive numbers up to 151 can be computed from the first 5 terms 1, 2, 4, 11, 34; e.g., 105 = (1 + 4) * (34 - 11 - 2). All terms are used, and each term is used only once. There is no such formula for 152, so a(6) = 152.
Programs
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Maple
restart: with(combinat): # generate numbers from set s GEN := proc(s) option remember; local erg,X,Y,x,y,i,a,b; if nops(s) < 2 then return s; fi; erg := {}; for i to nops(s)/2 do for a in choose(s,i) do b := s minus a; X := procname(a); Y := procname(b); for x in X do for y in Y do erg := erg union {x+y}; if x < y then erg := erg union {y-x}; elif x > y then erg := erg union {x-y}; fi; erg := erg union {x*y}; if type(x/y,integer) then erg := erg union {x/y}; elif type(y/x,integer) then erg := erg union {y/x}; fi; od; od; od; od; return erg; end: # minimal excluded number (not in set s) MEX := proc(s) local i; for i to infinity do if not member(i,s) then return i; fi; od; end: MaxIndex := 8; a := array(1..MaxIndex): w := {}: for n to MaxIndex do a[n] := MEX(GEN(w)); w := w union {a[n]}; od: seq(a[n],n=1..MaxIndex); -
Python
def a(n, v): R = dict() # index of each reachable subset is [card(s)-1][s] for i in range(n): R[i] = dict() for i in range(n): R[0][(v[i], )] = {v[i]} for j in range(1, n): for i in range((j+1)//2): for s1 in R[i]: for s2 in R[j-1-i]: if set(s1) & set(s2) == set(): s12 = tuple(sorted(set(s1) | set(s2))) if s12 not in R[len(s12)-1]: R[len(s12)-1][s12] = set() for a in R[i][s1]: for b in R[j-1-i][s2]: allowed = [a+b, a*b, a-b, b-a] if a!=0 and b%a==0: allowed.append(b//a) if b!=0 and a%b==0: allowed.append(a//b) R[len(s12)-1][s12].update(allowed) k = 1 while k in R[n-1][tuple(v)]: k += 1 return k alst = [1] [alst.append(a(n, alst)) for n in range(1, 8)] print(alst) # Michael S. Branicky, Oct 30 2022
Extensions
a(9) from Michael S. Branicky, Oct 30 2022
a(10) from Michael S. Branicky, Nov 07 2022
Comments