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 A056826 Primes p such that (p^p + 1)/(p + 1) is a prime. 5
 3, 5, 17, 157 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Note that (n^n+1)/(n+1) is prime only if n is prime, in which case it equals cyclotomic(2n,n), the 2n-th cyclotomic polynomial evaluated at x=n. This sequence is a subset of A088817. Are there only a finite number of these primes? - T. D. Noe, Oct 20 2003 (3^2 + 5^2)/2 = 17, (5^2 + 17^2)/2 = 157. - Thomas Ordowski, Jul 28 2013 Let b(1) = 1, b(2) = 3; b(n+2) = (b(n+1)^2 + b(n)^2)/2. Conjecture: if b(n) = p is prime then (p^p+1)/(p+1) is prime. Note that b(2) = 3, b(3) = 5, b(4) = 17, b(5) = 157 and b(10) is prime. - Thomas Ordowski, Jul 29 2013 Next term > 3000. - Seiichi Manyama, Mar 24 2018 No more terms through 6000. - Jon E. Schoenfield, Mar 25 2018 REFERENCES J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 157, p. 51, Ellipses, Paris 2008. R. K. Guy, Unsolved Problems in Theory of Numbers, 1994 A3. LINKS Eric Weisstein's World of Mathematics, Cyclotomic Polynomial MATHEMATICA Do[ If[ PrimeQ[ (Prime[ n ]^Prime[ n ] + 1)/(Prime[ n ] + 1) ], Print[ Prime[ n ] ] ], {n, 1, 213} ] Do[p=Prime[n]; If[PrimeQ[(p^p+1)/(p+1)], Print[p]], {n, 100}] (* T. D. Noe *) PROG (PARI) forprime(p=3, 1000, if(isprime((p^p+1)/(p+1)), print1(p", "))) \\ Seiichi Manyama, Mar 24 2018 CROSSREFS Cf. A088790 ((n^n-1)/(n-1) is prime), A088817 (cyclotomic(2n, n) is prime). Sequence in context: A107312 A083213 A171271 * A278138 A273870 A272060 Adjacent sequences:  A056823 A056824 A056825 * A056827 A056828 A056829 KEYWORD hard,nonn,more AUTHOR Robert G. Wilson v, Aug 29 2000 EXTENSIONS Definition corrected by Alexander Adamchuk, Nov 12 2006 STATUS approved

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Last modified October 15 17:24 EDT 2019. Contains 328037 sequences. (Running on oeis4.)