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A002822
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Numbers m such that 6m-1, 6m+1 are twin primes.
(Formerly M0641 N0235)
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90
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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
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OFFSET
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1,2
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COMMENTS
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6m-1 and 6m+1 are twin primes iff m is not of the form 6ab +- a +- b. - Jon Perry, Feb 01 2002
Even terms correspond to twin primes of the form (4k - 1, 4k + 1), odd terms to twin primes of the form (4k + 1, 4k + 3). - Lekraj Beedassy, Apr 03 2002
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)
A counterexample to "It appears that ...": Take j = 63. Then 6j = 378 and 36j = 2268. Now 379, 2267, and 2269 are prime, but 377 = 13 * 29. The sequence of counterexamples is A263282. - Jason Kimberley, Oct 13 2015
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 kp +- round(p/6) for positive integers k and primes p > 4, while twin ranks (this sequence) cannot be represented as kp +- round(p/6) for any k, p > 4. Here round(p/6) is the nearest integer to p/6. - Alexei Kourbatov, Jan 03 2015
Number of terms less than 10^k: 0, 5, 25, 142, 810, 5330, 37915, ... - Muniru A Asiru, Jan 24 2018
6m-1 and 6m+1 are twin primes iff 36m^2-1 is semiprime. It is algebraically provable that 36m^2-1 having any factor of the form 6k+-1 is equivalent to the statement that m is congruent to +-k (mod (6k+-1)). Other than the trivial case m=k, the fact of such a congruence means 36m^2-1 has a factor other than 6m-1 and 6m+1, and is not semiprime. Thus, {a(n)} lists the numbers m such that for all k < m, m is not congruent to +-k modulo (6k+-1). This is an alternative formulation of the results of Dinculescu referenced above. - Keith Backman, Apr 25 2021
Other than a(1)=1, it is provable that a(n) is not a square unless it is a multiple of 5, and a(n) is not a cube unless it is a multiple of 7. Examples of the former include a(11)=5^2=25, a(26)=10^2=100, and a(166)=35^2=1225; examples of the latter are rarer, including a(1531)=28^3=21952 and a(4163)=42^3=74088. - Keith Backman, Jun 26 2021
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REFERENCES
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W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 69.
W. Sierpiński, 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).
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LINKS
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FORMULA
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MAPLE
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select(n -> isprime(6*n-1) and isprime(6*n+1), [$1..1000]); # Robert Israel, Jan 11 2015
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MATHEMATICA
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Select[ Range[350], PrimeQ[6# - 1] && PrimeQ[6# + 1] & ]
Select[Range[400], AllTrue[6#+{1, -1}, PrimeQ]&] (* Harvey P. Dale, Jul 27 2022 *)
#/6&/@Select[Range[6, 2500, 6], AllTrue[#+{1, -1}, PrimeQ]&] (* Harvey P. Dale, Mar 31 2023 *)
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PROG
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(Magma) [n: n in [1..200] | IsPrime(6*n+1) and IsPrime(6*n-1)] // Vincenzo Librandi, Nov 21 2010
(PARI) p=5; forprime(q=5, 1e4, if(q-p==2, print1((p+1)/6", ")); p=q); \\ Altug Alkan, Oct 13 2015
(PARI) list(lim)=my(v=List(), p=5); forprime(q=7, 6*lim+1, if(q-p==2, listput(v, q\6)); p=q); Vec(v) \\ Charles R Greathouse IV, Dec 03 2016
(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
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001
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STATUS
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approved
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