A001039 a(n) = (p^p-1)/(p-1) where p = prime(n).

3, 13, 781, 137257, 28531167061, 25239592216021, 51702516367896047761, 109912203092239643840221, 949112181811268728834319677753, 91703076898614683377208150526107718802981

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Luis H. Gallardo, May 27 2022: (Start)

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.

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)

References

• S. Mattarei, The orders of nonsingular derivations of modular Lie algebras, Isr. J. Math., 132 (2002), 265-275.
• T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.
• T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
• 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.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..26
• J. Levine and R. E. Dalton, Minimum periods, modulo p, of first-order Bell exponential integers, Math. Comp., 16 (1962), 416-423.
• W. F. Lunnon, P. A. B. Pleasants, and N. M. Stephens, Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979), pp. 1-16. [From _N. J. A. Sloane_, Feb 07 2009]
• P. L. Montgomery, S. Nahm, and S. S. Wagstaff Jr, The period of the Bell numbers modulo a prime, Math. Comp. 79 (2010), 1793-1800.

Crossrefs

Cf. A054767, A214811.

Programs

Maple

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

Mathematica

Table[(Prime[n]^Prime[n] - 1)/(Prime[n] - 1), {n, 1, 10}]
(#^#-1)/(#-1)&/@Prime[Range[10]] (* Harvey P. Dale, Apr 09 2016 *)

Extensions

More terms from James Sellers, Jul 10 2000