OFFSET
1,2
COMMENTS
Numbers k such that A253236(k) = 0.
Numbers k such that all divisors of Phi_k(b) are congruent to 1 (mod k) 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 = 2*p^r is not in this sequence.
n = 3*p^r (p>3) is in this sequence iff p != 1 (mod 3).
n = 4*p^r (p>4) is in this sequence iff p != 1 (mod 4).
n = 5*p^r (p>5) is in this sequence iff p != 1 (mod 5).
...
n = k*p^r (p>k) is in this sequence iff p != 1 (mod k).
From Alex Eustis, Jan 06 2026: (Start)
Except for the first term 1, numbers k such that Phi_k(x) has no root mod q, where q is the largest prime divisor of k.
Except for the first term 1, numbers k such that r does not divide q-1, where q is the largest prime divisor of k, k = (q^m)*r, and q does not divide r. (End)
LINKS
Eric Chen and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Chen)
MATHEMATICA
q[k_] := Module[{p, e}, {p, e} = FactorInteger[k][[-1, ;; ]]; k ==1 || !Divisible[p-1, k/p^e]]; Select[Range[154], q] (* Amiram Eldar, Mar 15 2026 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric Chen, Apr 19 2015
STATUS
approved