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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

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^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 December 18 20:06 EST 2018. Contains 318245 sequences. (Running on oeis4.)