OIES sequence A002473 is the sequence of 7-smooth numbers: positive numbers whose prime divisors are all less than or equal to 7.
Let S be the infinite summation of all the reciprocals of the members of A002473. Does the sum converge, and if so then what is the sum?
Let {a} = all positive numbers whose divisors are at most 2
Let A = the infinite summation of A.
A = 1 + 1/2 + 1/2^2 + 1/2^3 + ... = 2/1 = 2
Let {b} = all positive numbers whose divisors are at most 3.
Then {a} is the subset of {b} including all elements which are not divisible by 3.
Dividing all elements of {b} by 3 gives a subset of {b} which is equal to {b} - {a}
Let B = the infinite summation of {b}.
Then, B/3 = B - A.
Substitute A = 2 and solve. B = 6/2 = 3
Let {c} = all positive numbers whose divisors are at most 5.
Then {b} is the subset of {c} including all elements which are not divisible by 5.
Dividing all elements of {c} by 5 gives a subset of {c} which is equal to {c} - {b}
Let C = the infinite summation of {c}.
Then, C/5 = C - B.
Substitute B = 3 and solve. C = 15/4
Let {d} = all positive numbers whose divisors are at most 7.
Then {c} is the subset of {d} including all elements which are not divisible by 7.
Dividing all elements of {d} by 7 gives a subset of {d} which is equal to {d} - {c}
Let D = the infinite summation of {d}.
Then, D/7 = D - C.
Substitute C = 15/4 and solve. D = 35/8
Final answer is 35/8. Fortunately, this agrees with Charlie's approximation.
Edited on January 23, 2019, 6:38 pm
Analytical solution (spoiler)
