An infinite sequence of positive integers is called a perfect sequence

An infinite sequence of positive integers is called a perfect
sequence if each term in the sequence is a perfect number,
that is, if each term can be expressed as the sum of its divisors,
excluding itself. For example, 6 is a perfect number, as its
divisors, 1, 2, and 3, sum to 6. Is the infinite sequence S a
perfect sequence?

(1) Exactly one term in S is a prime number.
(2) In sequence S, each term after the first in S has exactly 3 divisors.



The OA is D. I put down A as my answer. For statement 2 I get that a positive integer with 3 divisors will be the square of a prime number (4, 9, 25, 49, etc.). This statement though says that each term AFTER THE FIRST has 3 divisors. So the first term could be a perfect number or not. Maybe I'm missing something here. Any help on this will be helpful.

Hi Arbitraguer,

The question asks if S is a perfect sequence, meaning that every term must be perfect. (1) tells us the first term is not perfect; (2) tells us that no term after the first can be perfect. Therefore, each of (1) and (2) answers "NO" to the question of whether the entire sequence is perfect. Both are sufficient!

In the sequence S, if there is even one term which is not a perfect number, the sequence is not a perfect sequence. You need every term of the sequence to be a perfect number for the sequence to be a perfect sequence.
Statement 2 tells you that after the first term, every term is 'non-perfect'. We don't care whether the first term is perfect or not. Since we know that the sequence has non-perfect numbers, the sequence is not perfect. Hence, statement 2 is also sufficient.

Test makers like to add little twists like these "after the first term" to mess with your mind! I am sure you would have had no problems if the second statement were "...each term has exactly 3 divisors"

Indeed a very nice question !!!

Thanks again Bunnel for presenting solution with such a simplicity!!!

in statement 1 it is said that exactly one element is a prime number. There is no information about the rest of the numbers then how can take the set as prime? it is not sufficient ? i am i right?

For a sequence of positive integers to be a perfect sequence, it must consist exclusively of perfect numbers. Statement (1) indicates that one of the terms in sequence S is a prime number. As prime numbers are not perfect numbers, this means that not all terms in sequence S are perfect numbers. Consequently, sequence S does not meet the criteria for being a perfect sequence.

Hope it's clear.