

A056826


Primes p such that (p^p + 1)/(p + 1) is a prime.


5




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 2nth 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

Table of n, a(n) for n=1..4.
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^n1)/(n1) 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



