A307687 -id:A307687 - cp's OEIS frontend

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

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, 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, 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

Offset: 1

T. D. Noe, Mar 28 2006

nonn

Note that a(n)=1 iff n is a power of a prime.
Because every cyclotomic polynomial is irreducible, it is expected that a(n) exists for all n.
Note that if p=Phi(n,k) is prime and n>1, then p==1 (mod k). - Corrected by Robert Israel, Apr 22 2019

Jinyuan Wang, Table of n, a(n) for n = 1..5000 (terms 1..1000 from T. D. Noe)

Cf. A085398, A117545 (least k such that Phi(k, n) is prime), A307687.

f:= proc(n) local C, x, k;
  C:= unapply(numtheory:-cyclotomic(n, x), x);
  for k from 1 do if isprime(C(k)) then return k fi od
end proc:
map(f, [$1..200]); # Robert Israel, Apr 22 2019

Table[k=1; While[ !PrimeQ[Cyclotomic[n,k]], k++ ]; k, {n,100}]

a(n) = my(k=1); while (!isprime(polcyclo(n, k)), k++); k; \\ Michel Marcus, Apr 22 2019

Phi(n, a(n)) = A307687(n). - Robert Israel, Apr 22 2019

A085399 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the first prime Cn(x) after Cn(1).

Original entry on oeis.org

2, 3, 7, 5, 31, 3, 127, 17, 73, 11, 12207031, 13, 8191, 43, 151, 257, 131071, 46441, 524287, 61681, 368089, 683, 11111111111111111111111, 241, 705429635566498619547944801, 2731, 262657, 15790321, 7369130657357778596659, 331, 2147483647, 65537, 599479, 43691

Offset: 1

Don Reble, Jun 28 2003

nonn

			a(11)=12207031 because C11(x) is composite for x=2,3,4 and C11(5)=12207031 is prime.

Jinyuan Wang, Table of n, a(n) for n = 1..396

Cf. A084732, A085398, A307687.

a(n) = Phi(n,A085398(n)) where Phi(n,k) is the n-th cyclotomic polynomial evaluated at k. - Jinyuan Wang, Sep 04 2022

More terms from Jinyuan Wang, Sep 04 2022

cp's OEIS Frontend

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

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