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The tm_week(t, standard) function interprets its first argument t as a time in seconds from 1 Jan 1970. Despite the name of this function it does not report a field from the POSIX tm structure.
If standard = 0 it returns the week number in the ISO 8601 "week date" system. This corresponds to gnuplot's %W time format. If standard = 1 it returns the CDC epidemiological week number ("epi week"). This corresponds to gnuplot's %U time format. For corresponding inverse functions that convert week dates to calendar time see weekdate_iso, weekdate_cdc.
In brief, ISO Week 1 of year YYYY begins on the Monday closest to 1 Jan YYYY. This may place it in the previous calendar year. For example Tue 30 Dec 2008 has ISO week date 2009-W01-2 (2nd day of week 1 of 2009). Up to three days at the start of January may come before the Monday of ISO week 1; these days are assigned to the final week of the previous calendar year. E.g. Fri 1 Jan 2021 has ISO week date 2020-W53-5.
The US Center for Disease Control (CDC) epidemiological week is a similar week date convention that differs from the ISO standard by defining a week as starting on Sunday, rather than on Monday.