Consider the following sequence: https://oeis.org/A079523 Let $a(n)$ denote the terms of the sequence of utterly odd numbers. Show that for each $n$ there exists a $p$ such that for each $k\in \mathbb{N}$ we have that $a(n)+kp$ is also utterly odd.
There are some patterns here. Consider the $a(n)$ and the value of $p$ that works where $a(n)<100$:
For 1 the value of p is 4
For 5 the value of p is 4
For 7 the value of p is 16
For 9 the value of p is 4
For 13 the value of p is 4
For 17 the value of p is 4
For 21 the value of p is 4
For 23 the value of p is 16
For 25 the value of p is 4
For 29 the value of p is 4
For 31 the value of p is 64
For 33 the value of p is 4
For 37 the value of p is 4
For 39 the value of p is 16
For 41 the value of p is 4
For 45 the value of p is 4
For 49 the value of p is 4
For 53 the value of p is 4
For 55 the value of p is 16
For 57 the value of p is 4
For 61 the value of p is 4
For 65 the value of p is 4
For 69 the value of p is 4
For 71 the value of p is 16
For 73 the value of p is 4
For 77 the value of p is 4
For 81 the value of p is 4
For 85 the value of p is 4
For 87 the value of p is 16
For 89 the value of p is 4
For 93 the value of p is 4
For 95 the value of p is 64
For 97 the value of p is 4