Find this integer with these properties.

To find the smallest integer $n$ with these properties, it is enough to check that every positive integer up to $n-1$ does not satisfy all the conditions.

$1=2^0$ , condition 2

$2=2^1$ , condition 2

$3=3^1$ , condition 2

$4=3+1$ , condition 3

$5=5^1$ , condition 2

$6=5+1$ , condition 3

$7=7^1$ , condition 2

$8=2^3$ , condition 2

$9=3^2$ , condition 2

$10=1+2+3+4$ , condition 1

$11=11^1$ , condition 2

$12=11+1$ , condition 3

$13=13^1$ , condition 2

$14=13+1$ , condition 3

$15=1+2+3+4+5$ , condition 1

$16=2^4$ , condition 2

$17=17^1$ , condition 2

$18=17+1$ , condition 3

$19=19^1$ , condition 2

$20=19+1$ , condition 3

$21=1+2+3+4+5+6$ , condition 1

$22$ isn't the sum of the first k integers, because when $k=6$ the sum is $21$ and when $k=7$ the sum is $28$ . It's not a power of a prime, either, because it is the product of two distinct primes, and it's not $1$ more than a prime, because $21$ is composite.

The correct answer is $\boxed{22}$