Function that converts a vector of numbers to a vector of standard units

yes, scale():

x <- c(1,3,4,5,7)
scale(x)

The function you are looking for is scale.

scale(x)


           [,1]
[1,] -1.3416408
[2,] -0.4472136
[3,]  0.0000000
[4,]  0.4472136
[5,]  1.3416408
attr(,"scaled:center")
[1] 4
attr(,"scaled:scale")
[1] 2.236068

Note that the answers are not identical to what you posted in your question. The reason is that the standard deviation in your x is actually 2.23, not 2.

sd(x)
[1] 2.236068

How about simply (x-mean(x))/sd(x), or am I missing some subtlety here?