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Primes p such that k^p - (k-1)^p is composite: A087489 (k=3), A087490 (k=4), A087685 (k=5), A087749 (k=6), A087759 (k=7), this sequence (k=8), A087894 (k=9), A087895 (k=10).
Offset corrected by Mohammed Yaseen, Jul 18 2022
Cf. A016177.
Primes np such that 8^n-p - 7^np is composite.