A088817 -id:A088817 - cp's OEIS frontend

A070519 Numbers k such that Cyclotomic(k,k) (i.e., the value of k-th cyclotomic polynomial at k) is a prime number.

2, 3, 4, 6, 10, 12, 14, 19, 31, 46, 74, 75, 98, 102, 126, 180, 236, 310, 368, 1770, 1858, 3512, 4878, 5730, 7547, 7990, 8636, 9378, 11262

Offset: 1

Labos Elemer, May 02 2002

When n is prime, then the solutions are given in A088790.
No term of this sequence is congruent to 1 mod 4. In general, if k = s^2*t where t is squarefree and t == 1 (mod 4), then Cyclotomic(k,t*x^2) is the product of two polynomials. See the Wikipedia link below. - Jianing Song, Sep 25 2019
All terms <= 1858 have been proven with PARI's implementation of ECPP.  All larger terms are BPSW PRPs.  There are no further terms <= 30000. - Lucas A. Brown, Dec 28 2020

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Wikipedia, Aurifeuillean factorization

Cf. A070518, A070520, A088790 ((k^k-1)/(k-1) is prime), A088817 (cyclotomic(2k,k) is prime), A088875 (cyclotomic(k,-k) is prime).

Cf. A117544, A085398.

Do[s=Cyclotomic[n, n]; If[PrimeQ[s], Print[n]], {n, 2, 256}]

for(n=2,10^9,if(ispseudoprime(polcyclo(n,n)),print1(n,", "))); \\ Joerg Arndt, Jan 22 2015

More terms from T. D. Noe, Oct 17 2003

a(29) from Charles R Greathouse IV, May 05 2011

A056826 Primes p such that (p^p + 1)/(p + 1) is a prime.

Original entry on oeis.org

3, 5, 17, 157

Offset: 1

Robert G. Wilson v, Aug 29 2000

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(0) = 1, b(1) = 3; b(n+1) = (b(n)^2 + b(n-1)^2)/2. Conjecture: if b(n) = p is prime, then (p^p+1)/(p+1) is prime. Note that b(1) = 3, b(2) = 5, b(3) = 17, b(4) = 157 and b(9) is also 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

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.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A088790 ((n^n-1)/(n-1) is prime), A088817 (cyclotomic(2n, n) is prime).

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, Oct 20 2003 *)

forprime(p=3, 1000, if(isprime((p^p+1)/(p+1)), print1(p", "))) \\ Seiichi Manyama, Mar 24 2018

Definition corrected by Alexander Adamchuk, Nov 12 2006

A088875 Cyclotomic(n,-n) is prime.

Original entry on oeis.org

1, 3, 4, 5, 6, 9, 12, 14, 17, 82, 86, 157, 158, 180, 210, 236, 245, 368, 462, 842, 1034, 3512, 3977, 8636

Offset: 1

T. D. Noe, Oct 20 2003

This is a generalization of A056826. See A088817 for another generalization. Note that (n^n+1)/(n+1) = cyclotomic(n,-n) when n is prime. Also note that, for odd n>1, cyclotomic(n,-n) = cyclotomic(2n,n) and for n a multiple of 4, cyclotomic(n,-n) = cyclotomic(n,n).

Some of the larger entries may only correspond to probable primes.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A056826 ((n^n+1)/(n+1) is prime), A070519 (cyclotomic(n, n) is prime), A088817 (cyclotomic(2n, n) is prime).

Do[p=Prime[n]; If[PrimeQ[Cyclotomic[n, -n]], Print[p]], {n, 100}]

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A070519 Numbers k such that Cyclotomic(k,k) (i.e., the value of k-th cyclotomic polynomial at k) is a prime number.

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A056826 Primes p such that (p^p + 1)/(p + 1) is a prime.

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A088875 Cyclotomic(n,-n) is prime.

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