A006517 Numbers k such that k divides 2^k + 2.

1, 2, 6, 66, 946, 8646, 180246, 199606, 265826, 383846, 1234806, 3757426, 9880278, 14304466, 23612226, 27052806, 43091686, 63265474, 66154726, 69410706, 81517766, 106047766, 129773526, 130520566, 149497986, 184416166, 279383126

Offset: 1

Author

N. J. A. Sloane, David W. Wilson

Keywords

• nonn
• nice

Comments

All terms greater than 1 are even. If an odd term n>1 exists then n = m*2^k + 1 for some k>=1 and odd m. Then n divides 2^(m*2^k) + 1 and so does every prime factor p of n, implying that 2^(k+1) divides the multiplicative order of 2 modulo p and thus p-1. Therefore n = m*2^k + 1 is the product of prime factors of the form t*2^(k+1) + 1, implying that n-1 is divisible by 2^(k+1), a contradiction. - Max Alekseyev, Mar 16 2009

The sequence is infinite. In fact, its intersection with A055685 (given by A219037) is infinite (see Li et al. link). - Max Alekseyev, Oct 11 2012

All terms greater than 6 have at least three distinct prime factors. - Robert Israel, Aug 21 2014

References

• R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 142.
• W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #18
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Giovanni Resta, Table of n, a(n) for n = 1..150
• Kin Y. Li et al., Solution to Problem 323, Mathematical Excalibur 14(2), 2009, p. 3.
• V. Meally, Letter to N. J. A. Sloane, May 1975

Crossrefs

Cf. A006521, A015888, A015889, A015891, A015892, A015893, A015897, A015898, A015902, A015903, A015904, A015905, A015906.

Programs

Mathematica

Do[ If[ PowerMod[ 2, n, n ] + 2 == n, Print[n]], {n, 2, 1500000000, 4} ]
Join[{1},Select[Range[28*10^7],PowerMod[2,#,#]==#-2&]] (* Harvey P. Dale, Aug 13 2018 *)

PARI

is_A006517(n)=!(Mod(2,n)^n+2)  \\ M. F. Hasler, Oct 08 2012

Extensions

Corrected and extended by Joe K. Crump (joecr(AT)carolina.rr.com), Sep 12 2000 and Robert G. Wilson v, Sep 13 2000