Looking for songs with this numeric progression, or similar.
Manic Street Preachers - This Joke Sport Severed (verse only)
MGMT - Kids
One Republic - Apologize
Tommy Tutone - 867-5309 (Jenny)
“That Don’t Impress Me Much” by Shania Twain (verses)
“The Passenger” by Iggy Pop (half of the main riff is i-VI-III-VII, with the ending half ending on the V instead of the VII).
“The Other Side” by Red Hot Chili Peppers (verses)
“Play It All Night Long” by Warren Zevon
“Save Tonight” by Eagle Eye Cherry
“Radio Nowhere” by Bruce Springstreen (though the VII has the third in the bass instead of the root)
“Lucky Love” by Ace of Bass
“21 Guns” by Green Day
Just thought of another one…
“Listen To Your Heart” by Roxette (chorus)
What is numerical progression?
This progression is the minor flipside of the major progression vi-IV-I-V, which we had a thread about a few years ago. This time we just take the relative minor (the vi chord in a major key) to be the root. From the older thread there are some suggestions that in hindsight are probably in a minor key, so at least some of the following belong in this thread:
“Listen to Your Heart”, as already mentioned
“Alone” by Heart
“Zombie” by the Cranberries
“I Know Why The River Runs” by Julie Miller
“Building a Mystery” by Sarah MacLachlan
“Peace of Mind” by Boston
“Disarm” by Smashing Pumpkins (verse)
Thanks everyone, this is exciting! 
Songs with the same numeric (numerical? I don’t actually know) progression are not necessarily in the same key, but the chords have the same relationship to each other. For instance, if the chord progression is C major, G major, F major, C major in one song and G major, D major, C major, G major in another, the numeric(al) progression is the same.