A239564 -id:A239564 - cp's OEIS frontend

A239565 (Round(c^prime(n)) - 1)/prime(n), where c is the hexanacci constant (A118427).

6702, 23594, 301738, 14576792, 53653610, 2738173594, 38254296398, 143514673148, 2032676550562, 109797468019174, 6007838407290514, 22863415355711030, 1267938526864061370, 18523200405015238420, 70884650213591098558, 3989789924439684599434

Offset: 7

Vladimir Shevelev and Peter J. C. Moses, Mar 21 2014

nonn

For n>=7, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.

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)
Eric Weisstein's World of Mathematics, Hexanacci Constant

Cf. A007619, A007663, A238693, A238697, A238698, A238700, A118427, A239502, A239544, A239564.

A239566 (Round(c^prime(n)) - 1)/prime(n), where c is the heptanacci constant (A118428).

Original entry on oeis.org

7200, 25562, 332466, 16472758, 61145666, 3200477798, 45473543628, 172043098818, 2478186385762, 137291966046470, 7704742900338106, 29569459376703894, 1681851263230158754, 24987922624169214866, 96433670513455876108, 5566902760779797458210

Offset: 7

Peter J. C. Moses and Vladimir Shevelev, Mar 21 2014

nonn

For n>=7, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.

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)
Eric Weisstein's World of Mathematics, Heptanacci Constant

Cf. A007619, A007663, A238693, A238697, A238698, A238700, A239502, A239544, A239564, A239565.

All roots of the equation x^7-x^6-x^5-x^4-x^3-x^2-x-1 = 0
are the following: c=1.9919641966050350211,
-0.78418701799584451319 +/- 0.36004972226381653409*i,
-0.24065633852269642508 + /- 0.84919699909267892575*i,
  0.52886125821602342773 +/-  0.76534196109589443115*i.
Absolute values of all roots, except for septanacci constant c, are less than 1.
Conjecture. Absolute values of all roots of the equation x^n - x^(n-1) - ... -x - 1 = 0, except for n-bonacci constant c_n, are less than 1. If the conjecture is valid, then for sufficiently large k=k(n), for all m>=k, we have round(c_n^prime(m)) == 1 (mod 2*prime(m)) (cf. Shevelev link).

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).

Original entry on oeis.org

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

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

nonn

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.

			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.

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)

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

cp's OEIS Frontend

A239565 (Round(c^prime(n)) - 1)/prime(n), where c is the hexanacci constant (A118427).

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A239566 (Round(c^prime(n)) - 1)/prime(n), where c is the heptanacci constant (A118428).

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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).

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