login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002822 Numbers n such that 6n-1, 6n+1 are twin primes.
(Formerly M0641 N0235)
58
1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58, 70, 72, 77, 87, 95, 100, 103, 107, 110, 135, 137, 138, 143, 147, 170, 172, 175, 177, 182, 192, 205, 213, 215, 217, 220, 238, 242, 247, 248, 268, 270, 278, 283, 287, 298, 312, 313, 322, 325 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

6n-1 and 6n+1 are twin primes iff n is not of the form 6ab +- a +- b. - Jon Perry, Feb 01 2002

The above equivalence was rediscovered by Balestrieri, see link. - Charles R Greathouse IV, Jul 05 2011

Even entries correspond to twin primes of the form (4k - 1,4k + 1), odd entries to twin primes of the form (4k + 1,4k + 3). - Lekraj Beedassy, Apr 03 2002

A002822 U A067611 U A171696 = A001477. - Juri-Stepan Gerasimov, Feb 14 2010

From Bob Selcoe, Nov 28 2014: (Start)

Except for a(1)=1, all numbers in this sequence are congruent to (0, 2 or 3) mod 5.

It appears that when a(n)=6j, then j is also in the sequence (e.g., 138 = 6*23; 312 = 6*52).  This also appears to hold for sequence A191626. If true, then it suggests that when seeking large twin primes, good candidates might be 36*a(n) +- 1, n>=2.

Conjecture: There is at least one number in the sequence in the interval [5k, 7k] inclusive, k>=1. If true, then the twin prime conjecture also is true.

(End)

Dinculescu calls all terms in the sequence "twin ranks", and all other positive integers "non-ranks", see links. Non-ranks are given by the formula np +- [p/6] for natural n and prime p>4, while twin ranks (this sequence) cannot be represented as np +- [p/6] for any n, p>4. Here [p/6] is the nearest integer to p/6. - Alexei Kourbatov, Jan 03 2015

REFERENCES

W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 69.

W. SierpiƄski, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 120.

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..10000

F. Balestrieri, An Equivalent Problem To The Twin Prime Conjecture, arXiv:1106.6050v1 [math.GM]

A. Dinculescu, On Some Infinite Series Related to the Twin Primes, The Open Mathematics Journal, 5 (2012), 8-14.

A. Dinculescu, The Twin Primes Seen from a Different Perspective, The British Journal of Mathematics & Computer Science, 3 (2013), Issue 4, 691-698.

S. W. Golomb, Problem E969, Solution, Amer. Math. Monthly, 58 (1951), 338; 59 (1952), 44.

FORMULA

a(n) = A014574(n+1)/6. - Ivan N. Ianakiev, Aug 19 2013

MAPLE

select(n -> isprime(6*n-1) and isprime(6*n+1), [$1..1000]); # Robert Israel, Jan 11 2015

MATHEMATICA

Select[ Range[350], PrimeQ[6# - 1] && PrimeQ[6# + 1] & ]

PROG

(MAGMA) [n: n in [1..200] | IsPrime(6*n+1) and IsPrime(6*n-1)] // Vincenzo Librandi, Nov 21 2010

(PARI) select(primes(100), n->isprime(n-2)&&n>5)\6 \\ Charles R Greathouse IV, Jul 05 2011

(Haskell)

a002822 n = a002822_list !! (n-1)

a002822_list = f a000040_list where

   f (q:ps'@(p:ps)) | p > q + 2 || r > 0 = f ps'

                    | otherwise = y : f ps where (y, r) = divMod (q + 1) 6

-- Reinhard Zumkeller, Jul 13 2014

CROSSREFS

Complement of A067611.

Cf. A014574.

A191626 is a subsequence.

Sequence in context: A062442 A036964 A067162 * A191327 A109598 A117959

Adjacent sequences:  A002819 A002820 A002821 * A002823 A002824 A002825

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified May 27 22:20 EDT 2015. Contains 257884 sequences.