A253235 Numbers n such that the n-th cyclotomic polynomial has no root mod p for all primes p <= n.

1, 12, 15, 24, 28, 30, 33, 35, 36, 40, 44, 45, 48, 51, 56, 60, 63, 65, 66, 69, 70, 72, 75, 76, 77, 80, 84, 85, 87, 88, 90, 91, 92, 95, 96, 99, 102, 104, 105, 108, 112, 115, 117, 119, 120, 123, 124, 126, 130, 132, 133, 135, 138, 140, 141, 143, 144, 145, 150, 152, 153, 154

Offset: 1

Author

Eric Chen, Apr 19 2015

Keywords

• nonn

Comments

Numbers n such that A253236(n) = 0.

Numbers n such that all divisors of Phi_n(b) are congruent to 1 (mod n) for every natural number b.

If p is prime, k, r are natural numbers, then:

Every n = p^r is not in this sequence.

Every n = 2p^r is not in this sequence.

n = 3p^r (p>3) is in this sequence iff p != 1 (mod 3).

n = 4p^r (p>4) is in this sequence iff p != 1 (mod 4).

n = 5p^r (p>5) is in this sequence iff p != 1 (mod 5).

...

n = kp^r (p>k) is in this sequence iff p != 1 (mod k).

Links

• Eric Chen and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Chen)

Crossrefs

For A253236(n) = 2, 3, 5, 7, 11, 13, see A000079, A038754, A245478, A245479, A245480, A245481.

Programs

PARI

is(n)=my(P=polcyclo(n), f=factor(n)[, 1]); for(i=1, #f, if(#polrootsmod(P, f[i]), return(0))); 1 \\ Charles R Greathouse IV, Apr 20 2015