A239640 a(n) is the smallest number such that for n-bonacci constant c_n satisfies round(c_n^prime(m)) == 1 (mod 2*p_m) for every m>=a(n).

3, 3, 4, 5, 7, 7, 10, 13, 14, 14, 19, 23, 23, 31, 34, 34, 46, 50, 60, 65, 73, 79, 88, 92, 107, 113, 126, 139, 149, 168, 182, 198, 210, 227, 244, 265, 276, 292, 317, 340, 369, 384, 408, 436, 444, 480, 516, 540, 565, 606, 628, 669, 704, 735, 759, 810, 829, 895, 925

Offset: 2

Author

Vladimir Shevelev and Peter J. C. Moses, Mar 23 2014

Keywords

• nonn

Comments

The n-bonacci constant is a unique root x_1>1 of the equation x^n-x^(n-1)-...-x-1=0. So, for n=2 we have Fibonacci constant phi or golden ratio (A001622); for n=3 we have tribonacci constant (A058265); for n=4 we have tetranacci constant (A086088), for n=5 (A103814), for n=6 (A118427), etc.

Examples

			Let n=2, then c_2 = phi (Fibonacci constant). We have round(c_2^2)=3 is not == 1 (mod 4), round(c_2^3)=4 is not == 1 (mod 6), while round(c_2^5)=11 == 1 (mod 10) and one can prove that for p>=5, we have round(c_2^p) == 1 (mod 2*p). Since 5=prime(3), then a(2)=3.
		

Links

• S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
• V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)

Crossrefs

Cf. A001622, A007619, A007663, A058265, A086088, A103814, A118427, A118428, A238693, A238697, A238698, A238700, A239502, A239544, A239564, A239565, A239566.