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A056826
Primes p such that (p^p + 1)/(p + 1) is a prime.
5
3, 5, 17, 157
OFFSET
1,1
COMMENTS
Note that (k^k+1)/(k+1) is prime only if k is prime, in which case it equals cyclotomic(2k,k), the 2k-th cyclotomic polynomial evaluated at x=k. This sequence is a subsequence 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
No more terms through 20000. - Michael S. Branicky, Jul 30 2024
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 * A370879 A278138 A273870
KEYWORD
hard,nonn,more
AUTHOR
Robert G. Wilson v, Aug 29 2000
EXTENSIONS
Definition corrected by Alexander Adamchuk, Nov 12 2006
STATUS
approved