A001039 - OEIS

%I M2964 N1199 #46 Jul 18 2022 19:23:41

%S 3,13,781,137257,28531167061,25239592216021,51702516367896047761,

%T 109912203092239643840221,949112181811268728834319677753,

%U 91703076898614683377208150526107718802981

%N a(n) = (p^p-1)/(p-1) where p = prime(n).

%C From _Luis H. Gallardo_, May 27 2022: (Start)

%C Let r be a root of the trinomial x^p-x-1 in a fixed algebraic closure F of the finite field F_p. Radoux conjectured in 1975 (see References) that a(n) equals the multiplicative order of r in F. The conjecture seems still open.

%C Moreover, S. Mattarei proved in 2002 that there exists a finite-dimensional non-nilpotent Lie algebra of characteristic p which admits a nonsingular derivation of order a(n) if p is odd and of order 73 if p = 2. (End)

%D S. Mattarei, The orders of nonsingular derivations of modular Lie algebras, Isr. J. Math., 132 (2002), 265-275.

%D T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

%D T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.

%D C. Radoux, Nombres de Bell, modulo p premier, et extensions de degré p de F_p. C.R. Acad. Sci. Paris Ser. A-B, 281(21) (1975) A879-A882.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001039/b001039.txt">Table of n, a(n) for n = 1..26</a>

%H J. Levine and R. E. Dalton, <a href="http://dx.doi.org/10.1090/S0025-5718-1962-0148604-2">Minimum periods, modulo p, of first-order Bell exponential integers</a>, Math. Comp., 16 (1962), 416-423.

%H W. F. Lunnon, P. A. B. Pleasants, and N. M. Stephens, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa35/aa3511.pdf">Arithmetic properties of Bell numbers to a composite modulus I</a>, Acta Arithmetica 35 (1979), pp. 1-16. [From _N. J. A. Sloane_, Feb 07 2009]

%H P. L. Montgomery, S. Nahm, and S. S. Wagstaff Jr, <a href="http://dx.doi.org/10.1090/S0025-5718-10-02340-9">The period of the Bell numbers modulo a prime</a>, Math. Comp. 79 (2010), 1793-1800.

%p for i from 1 to 20 do printf(`%d,`,(ithprime(i)^ithprime(i) -1)/(ithprime(i)-1)) od:

%t Table[(Prime[n]^Prime[n] - 1)/(Prime[n] - 1), {n, 1, 10}]

%t (#^#-1)/(#-1)&/@Prime[Range[10]] (* _Harvey P. Dale_, Apr 09 2016 *)

%Y Cf. A054767, A214811.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Jul 10 2000