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 A077659 a(n) = smallest k>1 such that the sum n^k + (n+1)^k is prime, or -1 if no such k exists. 6
 2, 2, 4, 2, 2, 4, 2, 4, 2, 32 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Checking k up through 1024 suggests that the sequence may continue -1, 2, 4, 2, -1, 4, 2, -1, 2, -1, 16, 2, 8, 2, 2, 4, 4, -1, 2, 2, 4, 2, 4, 2, 2, 4, 4, 4, 2, ... For any a>1 and b>1, a^k + b^k is composite for all odd k>1. Hence if n^k + (n+1)^k is prime then k must be a power of 2. It is known that a(11) > 2^22. Is it possible that 11^2^m + 12^2^m is composite for all m > 0? LINKS EXAMPLE a(3)=4 because 3^2 + 4^2 = 25 is not prime, but 3^4 + 4^4 = 337 is prime. MATHEMATICA lst={}; For[n=1, n<=100, n++, k=2; While[k<=2^10 && !PrimeQ[n^k+(n+1)^k], k=2*k]; If[k<=2^10, AppendTo[lst, k], AppendTo[lst, -1]]]; lst CROSSREFS Cf. A078902. Cf. A080121. Sequence in context: A073103 A247257 A069177 * A212595 A087692 A093621 Adjacent sequences:  A077656 A077657 A077658 * A077660 A077661 A077662 KEYWORD hard,more,nonn AUTHOR T. D. Noe, Nov 14 2002 STATUS approved

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Last modified September 25 19:14 EDT 2020. Contains 337344 sequences. (Running on oeis4.)