IBM Ponder This #311 · March 2024

This problem was suggested by Latchezar Christov - thanks Latchezar!

Let $X_n$ be the sequence of $n$ natural numbers $a_0,a_1,\dots ,a_{n-1}$ for which three properties are true:

1. $a_i=a_{i-1}+i$ for $0<i<n$
2. None of the sequence terms is prime.
3. There is no other sequence with the above properties and a smaller initial term $a_0$.

Please note that because of property #1, there will only be n numbers in the sequence, for any given n.

The first such sequences are:

• $X_1=[1]$
• $X_2=[8,9]$
• $X_3=[9,10,12]$
• $X_4=[9,10,12,15]$
• $X_5=[15,16,18,21,25]$

Your Goal: Please find the initial term $a_0$ of the sequence $X_{1000}$

A Bonus "*" will be given for finding the initial term $a_0$ of the sequence $X_{2024}$

Solution

Best opened after a real attempt

To be added.