Weird maths problem

[Rhombic]

A friend forwarded me this problem that initially does not seem overly complicated to tackle but I have had a hard time with it and still no solution:

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Sequence A(n) is defined as follows:
with t=1, 
the following is done sequentially from i=1 to i=n:
if t is divisible by i: t+=i
else: t+=n
after the last iteration (i=n), A(n) is defined: A(n)=t

Prove that there exist infinitely many m such that A(m)<A(m-1)

To check initially that this seems like it is the case, I wrote a simple Python script:

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olda=1
n=0
listnumbers=[]
while n<300:
    a=1
    for i in range(n):
        if a%(i+1)==0:
            a+=(i+1)
        else:
            a+=n
    if olda>a:
        listnumbers.append(n)
    print(str(a))
    olda=a
    n+=1
print(listnumbers)

In this script, "listnumbers" is the list that contains all numbers m for which A(m)<A(m-1). As you can see, for the first 300 terms of sequence A, there are many. Running it for the first 3000 terms or so shows no sign of stopping.

I was thinking of trying to prove the opposite by induction (that there is a number Z such that for all m>Z, A(m)>(Am-1)) but it seems very complicated...

Any ideas?