cp's OEIS Frontend

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

%I A239566 #24 Feb 16 2025 08:33:21
%S A239566 7200,25562,332466,16472758,61145666,3200477798,45473543628,
%T A239566 172043098818,2478186385762,137291966046470,7704742900338106,
%U A239566 29569459376703894,1681851263230158754,24987922624169214866,96433670513455876108,5566902760779797458210
%N A239566 (Round(c^prime(n)) - 1)/prime(n), where c is the heptanacci constant (A118428).
%C A239566 For n>=7, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.
%H A239566 S. Litsyn and V. Shevelev, <a href="http://dx.doi.org/10.1142/S1793042105000339">Irrational Factors Satisfying the Little Fermat Theorem</a>, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
%H A239566 V. Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2014-March/012750.html">A property of n-bonacci constant</a>, Seqfan (Mar 23 2014)
%H A239566 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeptanacciConstant.html">Heptanacci Constant</a>
%F A239566 All roots of the equation x^7-x^6-x^5-x^4-x^3-x^2-x-1 = 0
%F A239566 are the following: c=1.9919641966050350211,
%F A239566 -0.78418701799584451319 +/- 0.36004972226381653409*i,
%F A239566 -0.24065633852269642508 + /- 0.84919699909267892575*i,
%F A239566   0.52886125821602342773 +/-  0.76534196109589443115*i.
%F A239566 Absolute values of all roots, except for septanacci constant c, are less than 1.
%F A239566 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).
%Y A239566 Cf. A007619, A007663, A238693, A238697, A238698, A238700, A239502, A239544, A239564, A239565.
%K A239566 nonn
%O A239566 7,1
%A A239566 _Peter J. C. Moses_ and _Vladimir Shevelev_, Mar 21 2014