You're searching for local maxima, essentially. Start off by normalizing your data to your target, i.e. 183, and search for values closest to zero. Those are your local maxima. I added data with values greater than your target to demonstrate.
df <- data.frame(DAYS = c(0,12,12,14,133,140,0,12,14,15,178,183,184,190,0,15,30,72,172,172.5))
df$localmin <- abs(df$DAYS - 183)
df
> df
DAYS localmin
1 0.0 183.0
2 12.0 171.0
3 12.0 171.0
4 14.0 169.0
5 133.0 50.0
6 140.0 43.0
7 0.0 183.0
8 12.0 171.0
9 14.0 169.0
10 15.0 168.0
11 178.0 5.0
12 183.0 0.0
13 184.0 1.0
14 190.0 7.0
15 0.0 183.0
16 15.0 168.0
17 30.0 153.0
18 72.0 111.0
19 172.0 11.0
20 172.5 10.5
targets <- which(diff(sign(diff(c(df$localmin, 183)))) == 2) + 1L
df$targets <- 0
df$targets[targets] <- 1
df
> df
DAYS localmin targets
1 0.0 183.0 0
2 12.0 171.0 0
3 12.0 171.0 0
4 14.0 169.0 0
5 133.0 50.0 0
6 140.0 43.0 1
7 0.0 183.0 0
8 12.0 171.0 0
9 14.0 169.0 0
10 15.0 168.0 0
11 178.0 5.0 0
12 183.0 0.0 1
13 184.0 1.0 0
14 190.0 7.0 0
15 0.0 183.0 0
16 15.0 168.0 0
17 30.0 153.0 0
18 72.0 111.0 0
19 172.0 11.0 0
20 172.5 10.5 1