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A253236 The unique prime p <= n such that n-th cyclotomic polynomial has a root mod p, or 0 if no such p exists.

%I A253236 #45 Jul 27 2020 14:34:53
%S A253236 0,2,3,2,5,3,7,2,3,5,11,0,13,7,0,2,17,3,19,5,7,11,23,0,5,13,3,0,29,0,
%T A253236 31,2,0,17,0,0,37,19,13,0,41,7,43,0,0,23,47,0,7,5,0,13,53,3,11,0,19,
%U A253236 29,59,0,61,31,0,2,0,0,67,17,0,0,71,0
%N A253236 The unique prime p <= n such that n-th cyclotomic polynomial has a root mod p, or 0 if no such p exists.
%C A253236 There is at most one prime p <= n such that n-th cyclotomic polynomial has a root mod p.
%C A253236 For prime n and every natural number k, a(n^k) = n.
%C A253236 If a(n) != 0, then a(n)|n.
%C A253236 a(n) is either 0 or A006530(n). See Corollary 23 of the Shevelev et al. link. - _Robert Israel_, Sep 07 2016
%H A253236 Eric Chen and Charles R Greathouse IV, <a href="/A253236/b253236.txt">Table of n, a(n) for n = 1..10000</a> (first 1500 terms from Chen)
%H A253236 V. Shevelev, G. Garcia-Pulgarin, J. M. Velasquez-Soto and J. H. Castillo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Castillo/castillo2.html">Overpseudoprimes, and Mersenne and Fermat Numbers as Primover Numbers</a>, J. of Integer Sequences, Vol. 15 (2012), Art. 12.7.7.
%p A253236 N:= 1000: # to get a(1) to a(N)
%p A253236 f:= proc(n) local p,x,C,v;
%p A253236     C:= numtheory:-cyclotomic(n,x);
%p A253236     p:= max(numtheory:-factorset(n));
%p A253236     for v from 0 to p-1 do
%p A253236       if eval(C,x=v) mod p = 0 then return p fi
%p A253236     od:
%p A253236     0
%p A253236 end proc:
%p A253236 f(1):= 0:
%p A253236 seq(f(n),n=1..N); # _Robert Israel_, Sep 07 2016
%t A253236 a[n_] := Module[{p, x, c, v}, c[x_] = Cyclotomic[n, x]; p = FactorInteger[ n][[-1, 1]]; For[v=0, v<p, v++, If[Mod[c[v], p] == 0, Return[p]]]; 0];
%t A253236 a[1] = 0;
%t A253236 Array[a, 100] (* _Jean-François Alcover_, Jul 27 2020, after Maple *)
%o A253236 (PARI) a(n) = forprime(p=2, n, if(#polrootsmod(polcyclo(n), p), return(p)))
%o A253236 (PARI) a(n)=my(P=polcyclo(n),f=factor(n)[,1]); for(i=1,#f, if(#polrootsmod(P, f[i]), return(f[i]))); 0 \\ _Charles R Greathouse IV_, Apr 07 2015
%Y A253236 For a(n) = 0, see A253235.
%Y A253236 For a(n) = 2, see A000079.
%Y A253236 For a(n) = 3, see A038754.
%Y A253236 For a(n) = 5, see A245478.
%Y A253236 For a(n) = 7, see A245479.
%Y A253236 For a(n) = 11, see A245480.
%Y A253236 For a(n) = 13, see A245481.
%Y A253236 Cf. A006530.
%K A253236 nonn
%O A253236 1,2
%A A253236 _Eric Chen_, Apr 07 2015