A085398 - OEIS

%I #120 Dec 21 2022 13:19:28

%S 3,2,2,2,2,2,2,2,2,2,5,2,2,2,2,2,2,6,2,4,3,2,10,2,22,2,2,4,6,2,2,2,2,

%T 2,14,3,61,2,10,2,14,2,15,25,11,2,5,5,2,6,30,11,24,7,7,2,5,7,19,3,2,2,

%U 3,30,2,9,46,85,2,3,3,3,11,16,59,7,2,2,22,2,21,61,41,7,2,2,8,5,2,2

%N Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.

%C Conjecture: a(n) is defined for all n. - _Eric Chen_, Nov 14 2014

%C Existence of a(n) is implied by Bunyakovsky's conjecture. - _Robert Israel_, Nov 13 2014

%H Jinyuan Wang, <a href="/A085398/b085398.txt">Table of n, a(n) for n = 1..5000</a> (terms 1..1500 from Eric Chen)

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bunyakovsky_conjecture">Bunyakowsky conjecture</a>

%F a(A072226(n)) = 2. - _Eric Chen_, Nov 14 2014

%F a(n) = A117544(n) except when n is a prime power, since if n is a prime power, then A117544(n) = 1. - _Eric Chen_, Nov 14 2014

%F a(prime(n)) = A066180(n), a(2*prime(n)) = A103795(n), a(2^n) = A056993(n-1), a(3^n) = A153438(n-1), a(2*3^n) = A246120(n-1), a(3*2^n) = A246119(n-1), a(6^n) = A246121(n-1), a(5^n) = A206418(n-1), a(6*A003586(n)) = A205506(n), a(10*A003592(n)) = A181980(n).

%e a(11) = 5 because C11(k) is composite for k = 2, 3, 4 and prime for k = 5.

%e a(37) = 61 because C37(k) is composite for k = 2, 3, 4, ..., 60 and prime for k = 61.

%p f:= proc(n) local k;

%p for k from 2 do if isprime(numtheory:-cyclotomic(n,k)) then return k fi od

%p end proc:

%p seq(f(n), n = 1 .. 100); # _Robert Israel_, Nov 13 2014

%t Table[k = 2; While[!PrimeQ[Cyclotomic[n, k]], k++]; k, {n, 300}] (* _Eric Chen_, Nov 14 2014 *)

%o (PARI) a(n) = k=2; while(!isprime(polcyclo(n, k)), k++); k; \\ _Michel Marcus_, Nov 13 2014

%Y Cf. A117544, A066180, A085399, A103795, A056993, A153438, A246119, A246120, A246121, A206418, A205506, A181980.

%Y Cf. A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A006314, A217071, A164989, A217072, A217073, A153440, A217074, A217075, A006313, A097475.

%K nonn

%O 1,1

%A _Don Reble_, Jun 28 2003