"the first day of the week" in 1 Corinthians 16:2

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I heard somewhere that in 1 Corinthians 16:2

Upon the first [day] of the week let every one of you lay by him in
store, as [God] hath prospered him, that there be no gatherings when I
come

There is some ambiguity about “the first day of the week”. I heard something like it could even be something like “the first week of each seven weeks” or so. Is it true? Can anyone, please, comment on it here?

EDIT:

I just found where I came across it – “From Pentecost to Prison” by Charles H. Welch, p. 128:

Acts 20:7 : “And upon the first day of the week, when the disciples
came together to break bread, Paul preached unto them, ready to depart
on the morrow; and continued his speech until midnight”

Much has been written regarding the true translation of the phrase
“The first day of the week”. The original reads En de te mia ton
sabbaton, and has sometimes been translated “The first of the
sabbaths”. The Companion Bible has the following note on this point:

"The first day of the week" – first day of the sabbaths, i.e. 
the first day for reckoning the seven sabbaths to Pentecost. 
It depended upon the harvest 

(Deut. 16:9: "Seven weeks shalt thou number unto thee: begin 
to number the seven weeks from [such time as] thou beginnest 
[to put] the sickle to the corn"), 

and was always from the morrow after the weekly sabbath when 
the wave sheaf was presented 

(Lev. 23:15: "And ye shall count unto you from the morrow 
after the sabbath, from the day that ye brought the sheaf of 
the wave offering; seven sabbaths shall be complete"). 

In John 20:1 

("The first [day] of the week cometh Mary Magdalene early, 
when it was yet dark, unto the sepulchre, and seeth the 
stone taken away from the sepulchre") 

this was the fourth day after the Crucifixion, "the Lord’s 
Passover". This was by Divine ordering. But in A. D. 57 it 
was twelve days after the week of unleavened bread, and 
therefore more than a fortnight later than in A.D. 29. 

The reader may feel that there is a weak point in this argument, for
there is no evidence given for the “twelve days” that this view
necessitates. Those who regard “the first day of the week” as
referring to Sunday, draw attention to the fact that if we assume this
day to be the Sabbath, then, as this day begins at sunset, by
traveling at day-break Paul would have been traveling on the Sabbath.
This would not have been likely in view of the Jews’ bitter opposition
to his teaching, and the apostle’s conciliatory attitude at Jerusalem
a few weeks later (Acts 21:21-24).

It is also a point worth considering that if we translate Sabbaton and
Sabbata as ‘Sabbath’ and ‘Sabbath day’, then there is no word for
‘week’ in the New Testament, which seems rather unlikely.