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A117544 Least k such that Phi(n,k), the n-th cyclotomic polynomial evaluated at k, is prime.

%I A117544 #24 Dec 17 2022 07:18:08
%S A117544 3,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,6,1,4,3,2,1,2,1,2,1,4,1,2,1,1,2,2,
%T A117544 14,3,1,2,10,2,1,2,1,25,11,2,1,5,1,6,30,11,1,7,7,2,5,7,1,3,1,2,3,1,2,
%U A117544 9,1,85,2,3,1,3,1,16,59,7,2,2,1,2,1,61,1,7,2,2,8,5,1,2,11,4,2,6,44,4,1,2,63
%N A117544 Least k such that Phi(n,k), the n-th cyclotomic polynomial evaluated at k, is prime.
%C A117544 Note that a(n)=1 iff n is a power of a prime.
%C A117544 Because every cyclotomic polynomial is irreducible, it is expected that a(n) exists for all n.
%C A117544 Note that if p=Phi(n,k) is prime and n>1, then p==1 (mod k). - Corrected by _Robert Israel_, Apr 22 2019
%H A117544 Jinyuan Wang, <a href="/A117544/b117544.txt">Table of n, a(n) for n = 1..5000</a> (terms 1..1000 from T. D. Noe)
%F A117544 Phi(n, a(n)) = A307687(n). - _Robert Israel_, Apr 22 2019
%p A117544 f:= proc(n) local C, x, k;
%p A117544   C:= unapply(numtheory:-cyclotomic(n, x), x);
%p A117544   for k from 1 do if isprime(C(k)) then return k fi od
%p A117544 end proc:
%p A117544 map(f, [$1..200]); # _Robert Israel_, Apr 22 2019
%t A117544 Table[k=1; While[ !PrimeQ[Cyclotomic[n,k]], k++ ]; k, {n,100}]
%o A117544 (PARI) a(n) = my(k=1); while (!isprime(polcyclo(n, k)), k++); k; \\ _Michel Marcus_, Apr 22 2019
%Y A117544 Cf. A085398, A117545 (least k such that Phi(k, n) is prime), A307687.
%K A117544 nonn
%O A117544 1,1
%A A117544 _T. D. Noe_, Mar 28 2006