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.

A193628 Integer part of A192851 / A191626.

Original entry on oeis.org

1, 4, 4, 3, 14, 21, 60, 69, 72, 101, 99, 98, 119, 95, 110, 93, 106, 112, 111, 92, 138, 144, 141, 120, 119, 132, 122, 109, 106, 98, 89, 106, 102, 102, 108, 91, 97, 95, 96, 102, 106, 122, 121, 121, 127, 142, 141, 139, 141, 132, 135, 134, 140, 141, 138, 137
Offset: 1

Views

Author

Andrea Raffetti, Aug 01 2011

Keywords

Comments

Conjecture: the sequence seems to converge asymptotically to 113.

Links

Crossrefs

Cf. A192851, A191626.

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

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

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

A192851 Integers n such that 6n, 36n, and 216n fall between pairs of twin primes, that is, 6n-1, 6n+1, 36n-1, 36n+1, 216n-1, and 216n+1 are prime.

Original entry on oeis.org

2, 12, 23, 45, 325, 703, 2705, 3598, 4218, 7338, 10698, 13562, 16478, 16665, 20195, 25195, 29678, 32312, 36228, 51882, 79628, 83522, 84513, 84525, 89453, 100028, 106710, 107712, 108868, 114527, 119142, 145590, 147758, 151557, 167155, 173960, 190547, 192588
Offset: 1

Views

Author

Andrea Raffetti, Jul 11 2011

Keywords

Comments

Infinite under Dickson's conjecture. - Charles R Greathouse IV, Jul 24 2011

Examples

			12 is in the list because 12*6=72, 12*36=432, 12*216=2592 are all between a pair of twin primes (71,73 and 431,433 and 2591,2593).

Links

Crossrefs

Subsequence of A191626 and hence A002822.
Cf. A014574.

Programs

  • Mathematica
    Select[Range[1000000], PrimeQ[6 # - 1] && PrimeQ[6 # + 1] && PrimeQ[36 # - 1] && PrimeQ[36 # + 1] && PrimeQ[216 # - 1] && PrimeQ[216 # + 1] &] (* T. D. Noe, Jul 26 2011 *)
Select[Range[193000],AllTrue[{6#-1,6#+1,36#-1,36#+1,216#-1,216#+1}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 21 2020 *)
  • PARI
    is(n)=isprime(6*n-1) && isprime(6*n+1) && isprime(36*n-1) && isprime(36*n+1) && isprime(216*n-1) && isprime(216*n+1) \\ Charles R Greathouse IV, Sep 15 2015
Showing 1-3 of 3 results.

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

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