Python 3, 8179 bytes
n=0
while 1:n+=1;a=[i for i in range(1,n+1)if n%i<1];n*len(a)/sum%sum(a)%1oror print(n)
Outputs indefinitely.
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while 1:n+=1;a=[i for i in range(1,n+1)if n%i<1];n*len(a)/sum%sum(a)%1oror print(n)
Outputs indefinitely.