A002822 -id:A002822 - cp's OEIS frontend

A027856 Dan numbers: numbers m of the form 2^j * 3^k such that m +- 1 are twin primes.

4, 6, 12, 18, 72, 108, 192, 432, 1152, 2592, 139968, 472392, 786432, 995328, 57395628, 63700992, 169869312, 4076863488, 10871635968, 2348273369088, 56358560858112, 79164837199872, 84537841287168, 150289495621632, 578415690713088, 1141260857376768

Offset: 1

Richard C. Schroeppel

nonn

Special twin prime averages (A014574).

Intersection of A014574 and A003586. - Jeppe Stig Nielsen, Sep 05 2017

			a(14) = 243*4096 = 995328 and {995327, 995329} are twin primes.

Ray Chandler, Table of n, a(n) for n = 1..62 (terms < 10^1000, first 55 terms from Donovan Johnson)

Cf. A014574, A060211, A078883, A078884.

Cf. also A002822, A033845, A058383, A003586.

Select[#, Total@ Boole@ Map[PrimeQ, # + {-1, 1}] == 2 &] &@ Select[Range[10^7], PowerMod[6, #, #] == 0 &] (* Michael De Vlieger, Dec 31 2016 *)
seq[max_] := Select[Sort[Flatten[Table[2^i*3^j, {i, 1, Floor[Log2[max]]}, {j, 0, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {-1, 1}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)

a(n) = A078883(n) + 1 = A078884(n) - 1. - Amiram Eldar, Aug 27 2024

Offset corrected by Donovan Johnson, Dec 02 2011

Entry revised by N. J. A. Sloane, Jan 01 2017

A063983 Least k such that k*2^n +/- 1 are twin primes.

Original entry on oeis.org

4, 2, 1, 9, 12, 6, 3, 9, 57, 30, 15, 99, 165, 90, 45, 24, 12, 6, 3, 69, 132, 66, 33, 486, 243, 324, 162, 81, 90, 45, 345, 681, 585, 375, 267, 426, 213, 429, 288, 144, 72, 36, 18, 9, 147, 810, 405, 354, 177, 1854, 927, 1125, 1197, 666, 333, 519, 1032, 516, 258, 129, 72

Offset: 0

Robert G. Wilson v, Sep 06 2001

nonn

Excluding the first three terms, all remaining terms have digital root 3, 6, or 9. - J. W. Helkenberg, Jul 24 2013

			a(3) = 9 because 9*2^3 = 72 and 71 and 73 are twin primes.
a(6) = 3 because 3*2^6 = 192 and {191, 193} are twin primes.
a(71) = 630 because 630*2^71 = 1487545442103938242314240 and {1487545442103938242314239, 1487545442103938242314241} are twin primes.

Richard Crandall and Carl Pomerance, 'Prime Numbers: A Computational Perspective,' Springer-Verlag, NY, 2001, page 12.

Pierre CAMI, Table of n, a(n) for n = 0..2300

Cf. A040040, A045753, A002822, A124065, A124518-A124522.
Cf. A071256, A060210, A060256. For records see A125848, A125019.
Cf. A076806 (requires odd k).

Table[Do[s=(2^j)*k; If[PrimeQ[s-1]&&PrimeQ[s+1],Print[{j,k}]], {k,1,2*j^2}],{j,0,100}]; (* outprint of a[j]=k *)
Do[ k = 1; While[ ! PrimeQ[ k*2^n + 1 ] || ! PrimeQ[ k*2^n - 1 ], k++ ]; Print[ k ], {n, 0, 50} ]
f[n_] := Block[{k = 1},While[Nand @@ PrimeQ[{-1, 1} + 2^n*k], k++ ];k];Table[f[n], {n, 0, 60}] (* Ray Chandler, Jan 09 2009 *)

More terms from Labos Elemer, May 24 2002

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A059960 Smaller term of a pair of twin primes such that prime factors of their average are only 2 and 3.

Original entry on oeis.org

5, 11, 17, 71, 107, 191, 431, 1151, 2591, 139967, 472391, 786431, 995327, 57395627, 63700991, 169869311, 4076863487, 10871635967, 2348273369087, 56358560858111, 79164837199871, 84537841287167, 150289495621631, 578415690713087, 1141260857376767

Offset: 1

Labos Elemer, Mar 02 2001

nonn

Lesser of twin primes p such that p+1 = (2^u)*(3^w), u,w >= 1.

Primes p(k) such that the number of distinct prime divisors of all composite numbers between p(k) and p(k+1) is 2. - Amarnath Murthy, Sep 26 2002

			a(11)+1 = 2*2*2*3*3*3*3*3*3*3*3*3*3 = 472392.

Ray Chandler, Table of n, a(n) for n = 1..61 (terms < 10^1000, first 49 terms from T. D. Noe)

Cf. A014574, A002822, A033845, A058383, A027856.
Cf. A052297, A075581, A075580, A075583, A075584, A075585, A075586, A075587, A075588, A075589.
Apart from initial terms, same as A078883.

nn=10^15; Sort[Reap[Do[n=2^i 3^j; If[n<=nn && PrimeQ[n-1] && PrimeQ[n+1], Sow[n-1]], {i, Log[2, nn]}, {j, Log[3, nn]}]][[2, 1]]]
Select[Select[Partition[Prime[Range[38*10^5]],2,1],#[[2]]-#[[1]]==2&][[All,1]],FactorInteger[#+1][[All,1]]=={2,3}&] (* The program generates the first 15 terms of the sequence. *)
seq[max_] := Select[Sort[Flatten[Table[2^i*3^j - 1, {i, 1, Floor[Log2[max]]}, {j, 1, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {0, 2}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)

a(n) = A027856(n+1) - 1. - Amiram Eldar, Mar 17 2025

A067611 Numbers of the form 6xy +- x +- y, where x, y are positive integers.

Original entry on oeis.org

4, 6, 8, 9, 11, 13, 14, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 29, 31, 34, 35, 36, 37, 39, 41, 42, 43, 44, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 93, 94

Offset: 1

Jon Perry, Feb 01 2002

nonn

Equivalently, numbers n such that either 6n-1 or 6n+1 is composite (or both are).
Numbers k such that 36*k^2 - 1 is not a product of twin primes. - Artur Jasinski, Dec 12 2007
Apart from initial zero, union of A046953 and A046954. - Reinhard Zumkeller, Jul 13 2014
From Bob Selcoe, Nov 18 2014: (Start)
Complementary sequence to A002822.
For all k >= 1, a(n) are the only positive numbers congruent to the following residue classes:
f == k (mod 6k+-1);
g == (5k-1) (mod 6k-1);
h == (5k+1) (mod 6k+1).
All numbers in classes g and h will be in this sequence; for class f, the quotient must be >= 1.
When determining which numbers are contained in this sequence, it is only necessary to evaluate f, g and h when the moduli are prime and the dividends are >= 2*k*(3*k - 1) (i.e., A033579(k)).
(End)
From Jason Kimberley, Oct 14 2015: (Start)
Numbers n such that A001222(A136017(n)) > 2.
The disjoint union of A060461, A121763, and A121765.
(End)
From Ralf Steiner, Aug 08 2018 (Start)
Conjecture 1: With u(k) = floor(k(k + 1)/4) one has A071538(a(u(k))*6) = a(u(k)) - u(k) + 1, for  k >= 2 (u > 1).
Conjecture 2: In the interval [T(k-1)+1, T(k)], with T(k) = A000217(k), k >= 2, there exists at least one number that is not a member of the present sequence. (End)
Also: numbers of the form n*p +- round(p/6) with some positive integer n and prime p >= 5. [Proof available on demand.] - M. F. Hasler, Jun 25 2019

			4 = 6ab - a - b with a = 1, b = 1.
6 = 6ab + a - b or 6ab - a + b with a = 1, b = 1.
5 cannot be obtained by any values of a and b in 6ab - a - b, 6ab - a + b, 6ab + a - b or 6ab + a + b.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. Balestrieri, An Equivalent Problem To The Twin Prime Conjecture, arXiv:1106.6050 [math.GM], 2011.

Cf. A002822, A010051, A037074, A046953, A046954, A060461, A070043, A070799, A121763, A121765, A136017, A136050, A071538 (pi_2).

Cf. A323674 (numbers 6xy +- x +- y including repetitions). - Sally Myers Moite, Jan 27 2019

Filtered([1..120], k-> not IsPrime(6*k-1) or not IsPrime(6*k+1)) # G. C. Greubel, Feb 21 2019

a067611 n = a067611_list !! (n-1)
a067611_list = map (`div` 6) $
   filter (\x -> a010051' (x-1) == 0 || a010051' (x+1) == 0) [6,12..]
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [1..100] | not IsPrime(6*n-1) or not IsPrime(6*n+1)]; // Vincenzo Librandi, Nov 19 2014

filter:= n -> not isprime(6*n+1) or not isprime(6*n-1):
select(filter, [$1..1000]); # Robert Israel, Nov 18 2014

Select[Range[100], !PrimeQ[6# - 1] || !PrimeQ[6# + 1] &]
Select[Range[100],AnyTrue[6#+{1,-1},CompositeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 05 2019 *)

for(n=1, 1e2, if(!isprime(6*n+1) || !isprime(6*n-1), print1(n", "))) \\ Altug Alkan, Nov 10 2015

[n for n in (1..120) if not is_prime(6*n-1) or not is_prime(6*n+1)] # G. C. Greubel, Feb 21 2019

Edited by Robert G. Wilson v, Feb 05 2002

Edited by Dean Hickerson, May 07 2002

A060461 Numbers k such that 6k-1 and 6k+1 are twin composites.

Original entry on oeis.org

20, 24, 31, 34, 36, 41, 48, 50, 54, 57, 69, 71, 79, 86, 88, 89, 92, 97, 104, 106, 111, 116, 119, 130, 132, 134, 136, 139, 141, 145, 149, 150, 154, 160, 167, 171, 174, 176, 179, 180, 189, 190, 191, 193, 196, 201, 207, 209, 211, 212, 219, 222, 223, 224, 225, 226

Offset: 1

Lekraj Beedassy, Apr 09 2001

nonn

A counterpart to A002822, which generates twin primes.
Intersection of A046953 and A046954. - Michel Marcus, Sep 27 2013
All terms can be expressed as (6ab+a+b OR 6cd-c-d) AND (6xy+x-y) for a,b,c,d,x,y positive integers. Example: 20=6*2*2-2-2 AND 20=6*3*1+3-1. - Pedro Caceres, Apr 21 2019

			a(9) = 57: the 9th twin composites among the odds are { 6*57-1, 6*57+1 }, i.e., (341, 343) or (11*31, 7^3).

Zak Seidov, Table of n, a(n) for n = 1..5000

Cf. A002822, A046953, A046954, A259826.

i=1:10000;
Q1 = 6*i-1;
Q2 = 6*i+1;
Q = union(Q1,Q2);
P = primes(max(Q));
AT = setxor(Q,P);
f = 0;
for j=1:numel(AT);
    K = AT(j);
    K2 = K+2;
    z = ismember(K2,AT);
    if z == 1;
        f = f+1;
        ATR(f,:) = K + 1;
    end
end
m6 = ATR./6;
% Jesse H. Crotts, Sep 05 2016

iscomp := proc(n) if n=1 or isprime(n) then RETURN(0) else RETURN(1) fi: end: for n from 1 to 500 do if iscomp(6*n-1)=1 and iscomp(6*n+1)=1 then printf(`%d,`,n) fi: od: # James Sellers, Apr 11 2001

Select[Range[200], !PrimeQ[6#-1]&&!PrimeQ[6#+1]&] (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)
Select[Range[300],AllTrue[6#+{1,-1},CompositeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 15 2015 *)
Select[Range@ 300, Times @@ Boole@ Map[CompositeQ, 6 # + {1, -1}] > 0 &] (* Michael De Vlieger, Sep 14 2016 *)

A060461()={my(maxx=5000); n=1; ctr=0; while(ctrBill McEachen, Apr 04 2015

from sympy import isprime; from sys import maxsize as oo
is_A060461 = lambda n: not (isprime(n*6-1) or isprime(n*6+1))
def A060461(n = None, first = oo, start = 1, end = oo):
    "Return the n-th term or a generator of up to 'first' terms less than 'end', starting at 'start'."
    if n: first = n
    seq = (m for m,_ in zip(filter(is_A060461, range(start,end)), range(first)))
    return max(seq) if n else seq
list(A060461(first=20)) # M. F. Hasler, Jul 10 2025

a(n) ~ n. More specifically, there are x - x/log x + O(x/log^2 x) terms of the sequence up to x. - Charles R Greathouse IV, Mar 03 2020

a(n) = A259826(n)/6. - M. F. Hasler, Jul 10 2025

More terms from James Sellers, Apr 11 2001

A045753 Numbers n such that 4n-1 and 4n+1 are both primes.

Original entry on oeis.org

1, 3, 15, 18, 27, 45, 48, 57, 60, 78, 87, 105, 108, 150, 165, 207, 255, 258, 273, 288, 330, 357, 363, 372, 402, 405, 417, 447, 468, 483, 507, 522, 528, 567, 585, 648, 672, 678, 750, 780, 792, 813, 825, 840, 843, 867, 882, 885, 918, 942, 963, 1005, 1023

Offset: 1

Felice Russo

			3 belongs to the sequence because 4*3+1 and 4*3-1 are both primes.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A040040, A002822, A124065, A124518-A124522, A063983.

[n: n in [1..2000] | IsPrime(4*n+1) and IsPrime(4*n-1)] // Vincenzo Librandi, Nov 18 2010

Select[Range[1023], And @@ PrimeQ[{-1, 1} + 4# ] &] (* Ray Chandler, Dec 06 2006 *)

list(lim)=my(v=List(),p=2); forprime(q=3,4*lim+1, if(q-p==2 && p%4==3, listput(v,q\4)); p=q); Vec(v) \\ Charles R Greathouse IV, Dec 03 2016

More terms from Erich Friedman

A124519 Numbers k such that 12k - 1 and 12k + 1 are twin primes.

Original entry on oeis.org

1, 5, 6, 9, 15, 16, 19, 20, 26, 29, 35, 36, 50, 55, 69, 85, 86, 91, 96, 110, 119, 121, 124, 134, 135, 139, 149, 156, 161, 169, 174, 176, 189, 195, 216, 224, 226, 250, 260, 264, 271, 275, 280, 281, 289, 294, 295, 306, 314, 321, 335, 341, 344, 355, 356, 379, 399

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

			1 is in the sequence since 12*1 - 1 = 11 and 12*1 + 1 = 13 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A040040, A045753, A002822, A124065, A124518, A124520, A124521, A124522, A063983.

Select[Range[400], And @@ PrimeQ[{-1, 1} + 12# ] &] (* Ray Chandler, Nov 16 2006 *)

Extended by Ray Chandler, Nov 16 2006

A124522 a(n) = smallest k such that 2nk-1 and 2nk+1 are primes.

Original entry on oeis.org

2, 1, 1, 9, 3, 1, 3, 12, 1, 3, 9, 3, 12, 15, 1, 6, 3, 2, 6, 6, 1, 15, 3, 4, 3, 6, 2, 48, 6, 1, 21, 3, 3, 15, 6, 1, 27, 3, 4, 3, 15, 5, 12, 15, 2, 9, 3, 2, 9, 6, 1, 3, 60, 1, 6, 24, 2, 3, 9, 2, 129, 12, 7, 9, 15, 5, 12, 27, 1, 3, 9, 3, 42, 45, 1, 90, 3, 2, 66, 21, 5, 63, 27, 16, 6, 6, 2, 12, 24, 1, 6

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A040040, A045753, A002822, A124065, A124518-A124522, A063983.

isA001359 := proc(n) RETURN( isprime(n) and isprime(n+2)) ; end: A124522 := proc(n) local k; k :=1 ; while true do if isA001359(2*n*k-1) then RETURN(k) ; fi ; k := k+1 ; od ; end: for n from 1 to 60 do printf("%d,",A124522(n)) ; od ; # R. J. Mathar, Nov 06 2006

f[n_] := Block[{k = 1},While[Nand @@ PrimeQ[{-1, 1} + 2n*k], k++ ];k];Table[f[n], {n, 91}] (* Ray Chandler, Nov 16 2006 *)
skp[n_]:=Module[{k=1},While[AnyTrue[2n k+{1,-1},CompositeQ],k++];k]; Join[{2},Array[skp,100,2]] (* Harvey P. Dale, Mar 30 2024 *)

{for(n=1,91,k=1;while(!isprime(2*n*k-1)||!isprime(2*n*k+1),k++);print1(k, ","))}

Edited and extended by Klaus Brockhaus and R. J. Mathar, Nov 06 2006

A056956 Numbers n such that 6n+1 and 6n+5 are both primes.

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 13, 16, 17, 18, 21, 27, 32, 37, 38, 46, 51, 52, 58, 63, 66, 73, 76, 77, 81, 83, 102, 107, 112, 123, 126, 128, 137, 142, 143, 146, 147, 151, 156, 161, 168, 181, 182, 202, 213, 216, 217, 237, 238, 241, 247, 248, 258, 261, 263, 266, 268, 277, 282

Offset: 1

Henry Bottomley, Jul 18 2000

nonn

Note that if prime p>3 then p mod 6 = 1 or 5.

			a(2)=2 since 6*2+1=13 and 6*2+5=17 are both prime.

M. F. Hasler, Table of n, a(n) for n = 1..5000

Cf. A002822, A024898, A024899, A059325.

Select[Range[300], And @@ PrimeQ /@ ({1, 5} + 6#) &] (* Ray Chandler, Jun 29 2008 *)

is(n)=isprime(n*6+1)&&isprime(n*6+5) \\ M. F. Hasler, Apr 05 2017

a(n) = (A023200(n+1)-1)/6 = (A046132(n+1)-5)/6 = A047847(n+1)/3

a(n) = floor(A087679(n+1)/6). - M. F. Hasler, Apr 05 2017

Edited by N. J. A. Sloane, Nov 07 2006

A046954 Numbers k such that 6*k + 1 is nonprime.

Original entry on oeis.org

0, 4, 8, 9, 14, 15, 19, 20, 22, 24, 28, 29, 31, 34, 36, 39, 41, 42, 43, 44, 48, 49, 50, 53, 54, 57, 59, 60, 64, 65, 67, 69, 71, 74, 75, 78, 79, 80, 82, 84, 85, 86, 88, 89, 92, 93, 94, 97, 98, 99, 104, 106, 108, 109, 111, 113, 114, 116, 117, 119, 120, 124, 127, 129, 130, 132, 133, 134, 136, 139, 140

Offset: 1

Felice Russo

nonn

Equals A171696 U A121763; A121765 U A171696 = A046953; A121763 U A121765 = A067611 where A067611 U A002822 U A171696 = A001477. - Juri-Stepan Gerasimov, Feb 13 2010, Feb 15 2010

These numbers (except 0) can be written as 6xy +-(x+y) for x > 0, y > 0. - Ron R Spencer, Aug 01 2016

			a(2)=8 because 6*8 + 1 = 49, which is composite.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A047845 (2n+1), A045751 (4n+1), A127260 (8n+1).

Cf. A046953, A008588, A016921, subsequence of A067611, complement of A024899.

Filtered([0..250], k-> not IsPrime(6*k+1)) # G. C. Greubel, Feb 21 2019

a046954 n = a046954_list !! (n-1)
a046954_list = map (`div` 6) $ filter ((== 0) . a010051' . (+ 1)) [0,6..]
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [0..250] | not IsPrime(6*n+1)]; // G. C. Greubel, Feb 21 2019

remove(k-> isprime(6*k+1), [$0..140])[]; # Muniru A Asiru, Feb 22 2019

a = Flatten[Table[If[PrimeQ[6*n + 1] == False, n, {}], {n, 0, 50}]] (* Roger L. Bagula, May 17 2007 *)
Select[Range[0, 200], !PrimeQ[6 # + 1] &] (* Vincenzo Librandi, Sep 27 2013 *)

is(n)=!isprime(6*n+1) \\ Charles R Greathouse IV, Aug 01 2016

[n for n in (0..250) if not is_prime(6*n+1)] # G. C. Greubel, Feb 21 2019

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar
Corrected by Juri-Stepan Gerasimov, Feb 13 2010, Feb 15 2010
Corrected by Vincenzo Librandi, Sep 27 2013

cp's OEIS Frontend

A027856 Dan numbers: numbers m of the form 2^j * 3^k such that m +- 1 are twin primes.

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A063983 Least k such that k*2^n +/- 1 are twin primes.

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A059960 Smaller term of a pair of twin primes such that prime factors of their average are only 2 and 3.

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A067611 Numbers of the form 6xy +- x +- y, where x, y are positive integers.

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A060461 Numbers k such that 6*k-1 and 6*k+1 are twin composites.

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A045753 Numbers n such that 4n-1 and 4n+1 are both primes.

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A060461 Numbers k such that 6k-1 and 6k+1 are twin composites.

A124519 Numbers k such that 12k - 1 and 12k + 1 are twin primes.