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A002822 Numbers n such that 6n-1, 6n+1 are twin primes.
(Formerly M0641 N0235)
57
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)

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]

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

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.

Equal to A014574(n)/6 for n>1.

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

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Last modified December 20 20:01 EST 2014. Contains 252289 sequences.