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A245478 Numbers k such that the k-th cyclotomic polynomial has a root mod 5.

%I A245478 #19 Feb 16 2025 08:33:23
%S A245478 1,2,4,5,10,20,25,50,100,125,250,500,625,1250,2500,3125,6250,12500,
%T A245478 15625,31250,62500,78125,156250,312500,390625,781250,1562500,1953125,
%U A245478 3906250,7812500,9765625,19531250,39062500,48828125,97656250,195312500,244140625,488281250
%N A245478 Numbers k such that the k-th cyclotomic polynomial has a root mod 5.
%C A245478 Numbers of the form 2^i * 5^j for 0 <= i <= 2 and j >= 0.
%D A245478 Trygve Nagell, Introduction to Number Theory. New York: Wiley, 1951, pp. 164-168.
%H A245478 Eric M. Schmidt, <a href="/A245478/b245478.txt">Table of n, a(n) for n = 1..500</a>
%H A245478 Eric Weisstein, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>.
%H A245478 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,5).
%F A245478 a(3j + i) = 2^(i-1)*5^j for i = 1,2,3 and j >= 0.
%F A245478 a(n) = 5*a(n-3). G.f.: -x*(4*x^2+2*x+1) / (5*x^3-1). - _Colin Barker_, Aug 01 2014
%e A245478 The 4th cyclotomic polynomial x^2 + 1 considered modulo 5 has a root x = 2, so 4 is in the sequence.
%o A245478 (Sage) def A245478(n) : return 2^((n-1)%3)*5^((n-1)//3)
%o A245478 (PARI) for(n=1,10^6,if(#polrootsmod(polcyclo(n),5),print1(n,", "))) /* by definition; rather inefficient. - _Joerg Arndt_, Jul 28 2014 */
%o A245478 (PARI) is(n)=n%8 && 2^valuation(n,2)*5^valuation(n,5)==n \\ _Charles R Greathouse IV_, Jul 29 2014
%o A245478 (PARI) Vec(-x*(4*x^2+2*x+1)/(5*x^3-1) + O(x^100)) \\ _Colin Barker_, Aug 01 2014
%Y A245478 Cf. A000079, A038754, A245479, A245480, A245481.
%K A245478 nonn,easy
%O A245478 1,2
%A A245478 _Eric M. Schmidt_, Jul 23 2014