# Find the next term

Here, we have a sequence.

0 , 15 , 23 , 155 , 287 , 539 , 767 , ?

Can you find the next element ?

HINT 1

These are related to primes.

HINT 2

There are three coins in each pocket!!!

HINT 3

2 + 3 = 5

• Is $155$ correct? :) – athin May 13 '19 at 4:17
• Are you asking about the fourth term in question? If so, then it is correct. – Ak19 May 13 '19 at 4:25
• Yes and ah ok.. Nice sequence as I'm still clueless till this point xD – athin May 13 '19 at 4:37

First, I define three functions:

- $$p(n)$$ is the n-th prime number
- $$s(n) = p(n) + p(n+1) + p(n+2)$$
in words: the sum of three consecutive prime numbers starting at prime number $$n$$
- $$a(n) = p(n) + p(n+1) - p(n+2)$$

Those functions produce the values

- $$p(n) = 2, 3, 5, 7, 11, 13, 17, 19, ...$$
- $$s(n) = 10, 15, 23, 31, 41, 49, 59, 71, ...$$
- $$a(n) = 0, 1, 1, 5, 7, 11, 13, 13, ...$$
for $$n=1,2,3,4,5,6,7,8, ...$$

Then we can express the sequence like that:

$$Sequence(n) = a(n) * s(n)$$

Giving

$$Sequence(1) = a(1) * s(1) = 0*10 = 0$$
$$Sequence(2) = a(2) * s(2) = 1*15 = 15$$
$$Sequence(3) = a(3) * s(3) = 1*23 = 23$$
$$Sequence(4) = a(4) * s(4) = 5*31 = 155$$
$$Sequence(5) = a(5) * s(5) = 7*41 = 287$$
$$Sequence(6) = a(6) * s(6) = 11*49 = 539$$
$$Sequence(7) = a(7) * s(7) = 13*59 = 767$$

and finally

$$Sequence(8) = a(8) * s(8) = 13*71 = 923$$

so the next element is

923

• It is correct, but there's a simple way to find the pattern!!! Try to find it ! ( +1) – Ak19 May 13 '19 at 9:38
• Interesting riddle! Tried for a while, couldn't guess the simpler pattern! Send me the answer, please! solomon.vimal@gmail.com – Solomon Vimal Jun 7 '19 at 2:43