A002822 Numbers m such that 6m-1, 6m+1 are twin primes.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

6m-1 and 6m+1 are twin primes iff m 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 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

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)

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

References

• 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).

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], 2011.
• 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.
• A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.
• S. W. Golomb, Problem E969, Solution, Amer. Math. Monthly, 58 (1951), 338; 59 (1952), 44.
• S. W. Golomb, Letter to N. J. A. Sloane, Mar 26 1984
• Matthew A. Myers, Comments on A002822, Letter to N. J. A. Sloane, Dec 04 2018

Crossrefs

Complement of A067611.

Intersection of A024898 and A024899.

A191626 is a subsequence.

Cf. A014574, A263282.

Programs

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

Magma

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

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] & ]
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 *)

PARI

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

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

Formula

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

Extensions

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