cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

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

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

A055685 Numbers k such that 2^k == -1 (mod k-1).

Original entry on oeis.org

2, 6, 66, 378, 1190, 1470, 25806, 58590, 134946, 137346, 170586, 272610, 285390, 420210, 538734, 592410, 618450, 680706, 778806, 1163066, 1520442, 1700946, 2099202, 2831010, 4020030, 4174170, 4516110, 5059890, 5215770
Offset: 1

Views

Author

Robert G. Wilson v, Jun 09 2000

Keywords

Comments

The sequence is infinite. In fact, even its intersection with A006517 (given by A219037) is infinite.

Links

Crossrefs

Cf. A064935, A006517, A219037, A296369.

Programs

  • Mathematica
    Do[If[PowerMod[2, n, n-1]==n-2, Print[n]], {n, 2, 12900000}]
  • Python
    A055685_list = [n for n in range(2,10**6) if pow(2,n,n-1) == n-2] # Chai Wah Wu, Nov 04 2019

Formula

a(n) = A296369(n) + 1.

Extensions

Edited by Max Alekseyev, Oct 11 2012
Incorrect term 4285390 removed by Chai Wah Wu, Nov 04 2019

A217468 Composite values of n such that 2^n == 2 (mod n*(n-1)).

Original entry on oeis.org

91625794219, 8796093022207, 1557609722332488343, 18216643597893471403
Offset: 1

Views

Author

V. Raman, Oct 04 2012

Keywords

Comments

No other terms below 2^64.
The next term is <= 25790417485109157029391019 = A019320(258). - Emmanuel Vantieghem, Dec 01 2014
Terms A019320(k) belongs to this sequence for k in A297412. - Max Alekseyev, Dec 29 2017

Examples

			a(1) = 91625794219 = (2^38 - 2^19 + 1)/3 = A019320(114).
a(2) = 8796093022207 = 2^43 - 1 = A019320(43).

Links

Crossrefs

Cf. A069051, A216822, A219037, A217465.
Intersection of A001567 and { 2*A014945(k) + 1 }.

Programs

Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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