I think the problem is that the week calculation somehow uses the first day of the year. I don't understand the internal mechanics, but you can see what I mean with this example: 
library(data.table)
dd <- seq(as.IDate("2013-12-20"), as.IDate("2014-01-20"), 1)
# dd <- seq(as.IDate("2013-12-01"), as.IDate("2014-03-31"), 1)
dt <- data.table(i = 1:length(dd),
                 day = dd,
                 weekday = weekdays(dd),
                 day_rounded = round(dd, "weeks"))
## Now let's add the weekdays for the "rounded" date
dt[ , weekday_rounded := weekdays(day_rounded)]
## This seems to make internal sense with the "week" calculation
dt[ , weeknumber := week(day)]
dt 
    i        day   weekday day_rounded weekday_rounded weeknumber
1:  1 2013-12-20    Friday  2013-12-17         Tuesday         51
2:  2 2013-12-21  Saturday  2013-12-17         Tuesday         51
3:  3 2013-12-22    Sunday  2013-12-17         Tuesday         51
4:  4 2013-12-23    Monday  2013-12-24         Tuesday         52
5:  5 2013-12-24   Tuesday  2013-12-24         Tuesday         52
6:  6 2013-12-25 Wednesday  2013-12-24         Tuesday         52
7:  7 2013-12-26  Thursday  2013-12-24         Tuesday         52
8:  8 2013-12-27    Friday  2013-12-24         Tuesday         52
9:  9 2013-12-28  Saturday  2013-12-24         Tuesday         52
10: 10 2013-12-29    Sunday  2013-12-24         Tuesday         52
11: 11 2013-12-30    Monday  2013-12-31         Tuesday         53
12: 12 2013-12-31   Tuesday  2013-12-31         Tuesday         53
13: 13 2014-01-01 Wednesday  2014-01-01       Wednesday          1
14: 14 2014-01-02  Thursday  2014-01-01       Wednesday          1
15: 15 2014-01-03    Friday  2014-01-01       Wednesday          1
16: 16 2014-01-04  Saturday  2014-01-01       Wednesday          1
17: 17 2014-01-05    Sunday  2014-01-01       Wednesday          1
18: 18 2014-01-06    Monday  2014-01-01       Wednesday          1
19: 19 2014-01-07   Tuesday  2014-01-08       Wednesday          2
20: 20 2014-01-08 Wednesday  2014-01-08       Wednesday          2
21: 21 2014-01-09  Thursday  2014-01-08       Wednesday          2
22: 22 2014-01-10    Friday  2014-01-08       Wednesday          2
23: 23 2014-01-11  Saturday  2014-01-08       Wednesday          2
24: 24 2014-01-12    Sunday  2014-01-08       Wednesday          2
25: 25 2014-01-13    Monday  2014-01-08       Wednesday          2
26: 26 2014-01-14   Tuesday  2014-01-15       Wednesday          3
27: 27 2014-01-15 Wednesday  2014-01-15       Wednesday          3
28: 28 2014-01-16  Thursday  2014-01-15       Wednesday          3
29: 29 2014-01-17    Friday  2014-01-15       Wednesday          3
30: 30 2014-01-18  Saturday  2014-01-15       Wednesday          3
31: 31 2014-01-19    Sunday  2014-01-15       Wednesday          3
32: 32 2014-01-20    Monday  2014-01-15       Wednesday          3
     i        day   weekday day_rounded weekday_rounded weeknumber
My workaround is this function:
https://github.com/geneorama/geneorama/blob/master/R/round_weeks.R
round_weeks <- function(x){
    require(data.table)
    dt <- data.table(i = 1:length(x),
                     day = x,
                     weekday = weekdays(x))
    offset <- data.table(weekday = c('Sunday', 'Monday', 'Tuesday', 'Wednesday', 
                                     'Thursday', 'Friday', 'Saturday'),
                         offset = -(0:6))
    dt <- merge(dt, offset, by="weekday")
    dt[ , day_adj := day + offset]
    setkey(dt, i)
    return(dt[ , day_adj])
}
Of course, you can easily change the offset to make Monday first or whatever.  The best way to do this would be to add an offset to the offset... but I haven't done that yet.
I provided a link to my simple geneorama package, but please don't rely on it too much because it's likely to change and not very documented.