OFFSET
1,6
COMMENTS
Normal operator precedence is followed, so multiplication and division are performed before addition or subtraction, and each operator only acts on the following term, so 2 / 3 * 4 equals (2 / 3) * 4.
Unlike A058377, which uses only addition and subtraction, this sequence has solutions for all values of n >= 3.
EXAMPLE
a(3) = 1 as 1 + 2 - 3 = 0 is the only solution.
a(4) = 1 as 1 - 2 - 3 + 4 = 0 is the only solution.
a(5) = 1 as 1 * 2 - 3 - 4 + 5 = 0 is the only solution. This is the first term where a solution exists while no corresponding solution exists in A058377.
a(6) = 5. The solutions, all of which use multiplication or division, are
1 + 2 * 3 + 4 - 5 - 6 = 0,
1 - 2 + 3 * 4 - 5 - 6 = 0,
1 - 2 * 3 + 4 - 5 + 6 = 0,
1 * 2 + 3 - 4 + 5 - 6 = 0,
1 - 2 / 3 / 4 - 5 / 6 = 0.
The last solution is the first that uses division.
a(7) = 8. Six solutions use just addition, division and multiplication. The other two are
1 + 2 - 3 * 4 * 5 / 6 + 7 = 0,
1 / 2 * 3 * 4 - 5 + 6 - 7 = 0.
a(15) = 6280. An example solution is
1 / 2 / 3 / 4 * 5 * 6 - 7 - 8 + 9 / 10 + 11 / 12 * 13 + 14 / 15 = 0
which includes four fractions that sum to 15, which is balanced by - 7 - 8.
a(20) = 611503. An example solution is
1 / 2 / 3 / 4 / 5 + 6 / 7 / 8 / 9 / 10 * 11 / 12 - 13 / 14 / 15 / 16
+ 17 / 18 - 19 / 20 = 0
which sums five fractions that include fourteen divisions.
MATHEMATICA
Table[Length@Select[Tuples[{"+", "-", "*", "/"}, k-1], ToExpression[""<>Riffle[ToString/@Range@k, #]]==0&], {k, 9}] (* Giorgos Kalogeropoulos, Apr 02 2021 *)
PROG
(Python)
from itertools import product
from fractions import Fraction
def a(n):
nn = ["Fraction("+str(i)+", 1)" for i in range(1, n+1)]
return sum(eval("".join([*sum(zip(nn, ops+("", )), ())])) == 0 for ops in product("+-*/", repeat=n-1))
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Apr 02 2021
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Scott R. Shannon, Mar 27 2021
STATUS
approved