A245478 Numbers k such that the k-th cyclotomic polynomial has a root mod 5.
1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500, 625, 1250, 2500, 3125, 6250, 12500, 15625, 31250, 62500, 78125, 156250, 312500, 390625, 781250, 1562500, 1953125, 3906250, 7812500, 9765625, 19531250, 39062500, 48828125, 97656250, 195312500, 244140625, 488281250
Offset: 1
Examples
The 4th cyclotomic polynomial x^2 + 1 considered modulo 5 has a root x = 2, so 4 is in the sequence.
References
- Trygve Nagell, Introduction to Number Theory. New York: Wiley, 1951, pp. 164-168.
Links
- Eric M. Schmidt, Table of n, a(n) for n = 1..500
- Eric Weisstein, Cyclotomic Polynomial.
- Index entries for linear recurrences with constant coefficients, signature (0,0,5).
Programs
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PARI
for(n=1,10^6,if(#polrootsmod(polcyclo(n),5),print1(n,", "))) /* by definition; rather inefficient. - Joerg Arndt, Jul 28 2014 */
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PARI
is(n)=n%8 && 2^valuation(n,2)*5^valuation(n,5)==n \\ Charles R Greathouse IV, Jul 29 2014
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PARI
Vec(-x*(4*x^2+2*x+1)/(5*x^3-1) + O(x^100)) \\ Colin Barker, Aug 01 2014
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Sage
def A245478(n) : return 2^((n-1)%3)*5^((n-1)//3)
Formula
a(3j + i) = 2^(i-1)*5^j for i = 1,2,3 and j >= 0.
a(n) = 5*a(n-3). G.f.: -x*(4*x^2+2*x+1) / (5*x^3-1). - Colin Barker, Aug 01 2014
Comments