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

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

Offset: 7

Author

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

Keywords

• nonn

Comments

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

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)
• Eric Weisstein's World of Mathematics, Heptanacci Constant

Crossrefs

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

Formula

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).