Find the next term

Question

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

Answer 1

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