A060461 -id:A060461 - cp's OEIS frontend

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

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

A171696 Nonnegative numbers k such that neither of 6*k +- 1 is prime.

Original entry on oeis.org

0, 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

Offset: 1

Juri-Stepan Gerasimov, Dec 15 2009

nonn

A060461 is the main sequence for this entry.

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

Cf. A001477.

The same as A060461, except for the initial 0.

isA171696 := proc(n) not isprime(6*n+1) and not isprime(6*n-1) ; end proc: for n from 0 to 300 do if isA171696(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Mar 18 2010

Select[Range[0, 225], !Or @@ PrimeQ[6# + {-1, 1}] &] (* Amiram Eldar, Jan 21 2020 *)

is(n)=!isprime(6*n-1) && !isprime(6*n+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n+1) = A060461(n).

a(n) = n + O(n/log n). - Charles R Greathouse IV, Feb 17 2017

A121765 Numbers n such that 6n-1 is composite while 6n+1 is prime.

Original entry on oeis.org

6, 11, 13, 16, 21, 26, 27, 35, 37, 46, 51, 55, 56, 61, 62, 63, 66, 68, 73, 76, 81, 83, 90, 91, 96, 101, 102, 105, 112, 115, 118, 121, 122, 123, 125, 126, 128, 131, 142, 146, 151, 153, 156, 161, 165, 166, 168, 173, 178, 181, 186, 187, 188, 195, 200, 202, 206, 208, 216

Offset: 1

Lekraj Beedassy, Aug 20 2006

nonn

Entries in A046953 which are not in A060461 or equivalently, entries in A024899 which are not in A002822.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A121764.

Filtered([1..250], k-> not IsPrime(6*k-1) and IsPrime(6*k+1)); # G. C. Greubel, Feb 20 2019

[n: n in [1..250] | not IsPrime(6*n-1) and  IsPrime(6*n+1)]; // G. C. Greubel, Feb 20 2019

Select[Range[250], ! PrimeQ[6# - 1] && PrimeQ[6# + 1] &] (* Ray Chandler, Aug 22 2006 *)

for(n=1, 250, if(!isprime(6*n-1) && isprime(6*n+1), print1(n", "))) \\ G. C. Greubel, Feb 20 2019

[n for n in (1..250) if not is_prime(6*n-1) and  is_prime(6*n+1)] # G. C. Greubel, Feb 20 2019

Extended by Ray Chandler, Aug 22 2006

A121763 Numbers n such that 6n-1 is prime while 6n+1 is composite.

Original entry on oeis.org

4, 8, 9, 14, 15, 19, 22, 28, 29, 39, 42, 43, 44, 49, 53, 59, 60, 64, 65, 67, 74, 75, 78, 80, 82, 84, 85, 93, 94, 98, 99, 108, 109, 113, 114, 117, 120, 124, 127, 129, 133, 140, 144, 148, 152, 155, 157, 158, 159, 162, 163, 164, 169, 183, 184, 185, 194, 197, 198, 199

Offset: 1

Lekraj Beedassy, Aug 20 2006

nonn

Entries of A024898 which are not in A002822 or equivalently, entries of A046954 which are not in A060461.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A121762, A121765.

Filtered([1..250], k-> IsPrime(6*k-1) and not IsPrime(6*k+1)); # G. C. Greubel, Feb 20 2019

[n: n in [1..250] | IsPrime(6*n-1) and not IsPrime(6*n+1)]; // G. C. Greubel, Feb 20 2019

Select[Range[200], PrimeQ[6# -1] && !PrimeQ[6# +1] &] (* Ray Chandler, Aug 22 2006 *)

for(n=1, 250, if(isprime(6*n-1) && !isprime(6*n+1), print1(n", "))) \\ G. C. Greubel, Feb 20 2019

[n for n in (1..250) if is_prime(6*n-1) and not is_prime(6*n+1)] # G. C. Greubel, Feb 20 2019

Extended by Ray Chandler, Aug 22 2006

A171697 1 together with pairs of composites of the form (6n-1, 6n+1).

Original entry on oeis.org

1, 119, 121, 143, 145, 185, 187, 203, 205, 215, 217, 245, 247, 287, 289, 299, 301, 323, 325, 341, 343, 413, 415, 425, 427, 473, 475, 515, 517, 527, 529, 533, 535, 551, 553, 581, 583, 623, 625, 635, 637, 665, 667, 695, 697, 713, 715, 779, 781, 791, 793, 803

Offset: 1

Juri-Stepan Gerasimov, Dec 15 2009

Cf. A001477, A060461, A171696.

A157966 Number of 3's in A157733(n).

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 5, 5, 7, 9, 9, 11, 13, 13, 15, 17, 19, 19, 21, 23, 23, 25, 27, 29, 31, 33, 33, 35, 35, 37, 41, 43, 45, 45, 49, 49, 51, 53, 55, 57, 59, 59, 63, 63, 65, 65, 69, 73, 75, 75, 77, 79, 79, 83, 85, 87, 89, 89, 91, 93, 93, 97, 101, 103, 103, 105, 109, 111, 115, 115, 117

Offset: 0

Paul Curtz, Mar 10 2009

Odd numbers missing in the sequence: 39, 47, 61, 67, 71, 81, 95, 99, 107, 113, 137, 141 etc.

The numbers above are essentially 2*A060461(n) - 1. - Mikhail Kurkov, Dec 18 2021

Cf. A157733, A060461.

a(n) = if(n==2,1, (prime(n)-2)\3); \\ Kevin Ryde, Apr 25 2024

a(n) = floor((prime(n)-2)/3) for n != 2. - Kevin Ryde, Apr 25 2024

Edited and extended by R. J. Mathar, Mar 15 2009

A259826 Numbers n such that n is a multiple of 6 and both n-1 and n+1 are composite.

Original entry on oeis.org

120, 144, 186, 204, 216, 246, 288, 300, 324, 342, 414, 426, 474, 516, 528, 534, 552, 582, 624, 636, 666, 696, 714, 780, 792, 804, 816, 834, 846, 870, 894, 900, 924, 960, 1002, 1026, 1044, 1056, 1074, 1080, 1134, 1140, 1146, 1158, 1176, 1206, 1242, 1254, 1266, 1272, 1314, 1332, 1338, 1344, 1350

Offset: 1

Antonio Gimenez, Jul 05 2015

nonn

From Brian Almond, Jun 23 2020: (Start)
For every prime gap g, there is a run of consecutive a(n) of length max{[(g+2)/6]-1,0}.
Gaps between successive a(n) correspond to clusters of primes all within +- 8 of each other.  The number of primes within a gap G = a(n+1) - a(n) ranges from (G/6 - 1) to (G/6 - 1) plus the number of twin primes within the gap.
Record gaps in a(n) are 24 at a(1)=120, 42 at a(2)=144, 72 at a(10)=342 and 84 at a(1003)=14706 (the next gaps of 84 occur at a(43136164)=369008652 and a(643519601)=5244999552).  No larger record gaps exist below 10^10 (n <= 1239026836).
(End)
Define a "small-gap k-tuple" to be an admissible k-tuple with all of its gaps in {2,4,6,8}.  Every gap G = a(n+1) - a(n) >= 18 contains a small-gap k-tuple with k >= G/6 - 1 and diameter G-14, G-12 or G-10.  For example, at n=40 the gap between 1080 and 1134 contains the 9-tuple p+{0,4,6,10,16,22,30,36,42} for p=1087. - Brian Almond, Jul 25 2020

			For n=120, 120 is a multiple of 6, and both 119 and 121 are composite.

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Intersection of A008588 and A099047. - Michel Marcus, Jul 06 2015

Cf. A060461.

[n: n in [6..2000 by 6] | not IsPrime(n-1) and not IsPrime(n+1)]; // Vincenzo Librandi, Jul 08 2015

Select[6*Range[500], AllTrue[# + {1, -1}, CompositeQ] &] (* Harvey P. Dale, May 21 2017 *)

select(x->!isprime(x-1)&&!isprime(x+1), vector(10^3,j,6*j) ) \\ Joerg Arndt, Jul 06 2015

a(n) = 6 * A060461(n). - Brian Almond, Jun 22 2020

A384102 Least x in absolute value, such that there exists y, |x| >= |y| > 0, such that n = |6xy + x + y|, or 0 if no such x exists. Choose x > 0 if x and -x are both possible.

Original entry on oeis.org

0, 0, 0, -1, 0, 1, 0, 1, -2, 0, 2, 0, -2, -3, 2, 3, 0, 0, -4, -2, 4, 3, 0, 2, 0, 5, -4, 2, 4, 0, -3, 0, 0, -5, 3, 5, -3, 0, -8, 0, 3, -4, 6, -9, 0, 4, 0, -3, -10, -4, 10, 0, -5, 3, -8, 11, 5, 0, -12, 3, 12, -9, -5, -6, -4, 13, 5, 6, -10, 0, 4, 0, -4, -15, -7, -6, 0, 11, 4, 6, 16, -5, -12, -17, 12, -8, 0, -4, -7, 8, 18, -5, 7, -19, 0, 4, -9, 5, -6

Offset: 1

M. F. Hasler, Jun 20 2025

(6n-1, 6n+1) are twin primes iff a(n) = 0, that is, if there are no nonzero integers x, y such that n = |6xy + x + y|. (These n are listed in A002822, the complement is A067611.)

a(n) <= (6*n-1)/5, with equality if 6*n+1 is prime and 6*n-1 is 5 times a prime. - Robert Israel, Jul 21 2025

			For n = 1, 2 and 3, there are no nonzero x,y such that n = |6xy + x + y|, and (6n-1, 6n+1) = (5, 7), (11, 13) and (17, 19) are indeed twin primes.
For n = 4 we have x = y = -1 such that |6xy + x + y| = |6 - 1 - 1| = 4 and (23, 25) is indeed not a twin prime pair.

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

Cf. A384103 (the corresponding y-values).
Cf. A002822 (indices of zeros: n such that 6n-1 and 6n+1 are twin primes).
Cf. A077800 (list of twin primes), A060461, A171696 (none among 6n+-1 is prime), A067611 (n = 6xy +- x +- y: 6n-1 or 6n+1 is composite).

f:= proc(n) local V, C, t, m,v, r;
       V:= numtheory:-divisors(6*n+1) minus {1,6*n+1};
       C:= map(u -> `if`(u mod 6 = 1,  [(u-1)/6, ((6*n+1)/u - 1)/6], [(-u-1)/6, (-(6*n+1)/u - 1)/6]), V);
       V:= numtheory:-divisors(6*n-1) minus {1,6*n-1};
       C:= C union map(u -> `if`(u mod 6 = 1, [(u-1)/6, ((-6*n+1)/u - 1)/6], [(-u-1)/6, ((-6*n+1)/u - 1)/6]), V);
       C:= select(t -> abs(t[1]) >= abs(t[2]), C)[..,1];
       if C = {} then return 0 fi;
       m:= infinity;
       for t in C do
         if abs(t) < m then m:= abs(t); r:= t;
         elif abs(t) = m and t > 0 then r:= t
         fi
       od;
       r
 end proc:
map(f, [$1..100]); # Robert Israel, Jul 21 2025

{A384102(n)=for(x=1,n\/5, my(p=6*x+1, q=6*x-1, r=if((n-x)%p==0, (n-x)\p, (n+x)%p==0, (n+x)\p, (n-x)%q==0, (x-n)\q, (n+x)%q==0,-(n+x)\q)); r && abs(r) <= x && return(sign(r)*x))}

A384103 a(n) = y with minimum |x| >= |y| > 0, such that n = |6xy + x + y|, or 0 if no such x, y exist. If x and -x are solutions, choose x > 0 > y = -x.

Original entry on oeis.org

0, 0, 0, -1, 0, -1, 0, 1, -1, 0, -1, 0, 1, -1, 1, -1, 0, 0, -1, -2, -1, 1, 0, -2, 0, -1, 1, 2, 1, 0, -2, 0, 0, 1, -2, 1, 2, 0, -1, 0, 2, -2, 1, -1, 0, -2, 0, -3, -1, 2, -1, 0, -2, -3, 1, -1, -2, 0, -1, 3, -1, 1, 2, -2, -3, -1, 2, -2, 1, 0, -3, 0, 3, -1, -2, 2, 0, 1, 3, 2, -1, -3, 1, -1, 1, -2, 0, -4, 2, -2

Offset: 1

M. F. Hasler, Jun 20 2025

(6n-1, 6n+1) are twin primes iff a(n) = 0, that is, if there are no nonzero integers x, y such that n = |6xy + x + y|. These n are listed in A002822, the complement is A067611.

The corresponding x-values are listed in A384102.

			For n = 1, 2 and 3, there are no nonzero x,y such that n = |6xy + x + y|, and (6n-1, 6n+1) = (5, 7), (11, 13) and (17, 19) are indeed twin primes.
For n = 4 we have x = y = -1 such that |6xy + x + y| = |6 - 1 - 1| = 4 and (23, 25) is indeed not a twin prime pair.

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

Cf. A384102 (the corresponding x-values).
Cf. A002822 (indices of zeros: n such that 6n-1 and 6n+1 are twin primes).
Cf. A077800 (list of twin primes), A060461, A171696 (none among 6n+-1 is prime), A067611 (n = 6xy +- x +- y: 6n-1 or 6n+1 is composite).

f:= proc(n) local V, C, t, m, v, r;
       V:= numtheory:-divisors(6*n+1) minus {1, 6*n+1};
       C:= map(u -> `if`(u mod 6 = 1,  [(u-1)/6, ((6*n+1)/u - 1)/6], [(-u-1)/6, (-(6*n+1)/u - 1)/6]), V);
       V:= numtheory:-divisors(6*n-1) minus {1, 6*n-1};
       C:= C union map(u -> `if`(u mod 6 = 1, [(u-1)/6, ((-6*n+1)/u - 1)/6], [(-u-1)/6, ((6*n-1)/u - 1)/6]), V);
       C:= select(t -> abs(t[1]) >= abs(t[2]), C);
       if C = {} then return 0 fi;
       m:= infinity;
       for t in C do
         if abs(t[1]) < m then m:= abs(t[1]); r:= t[2];
         elif abs(t[1]) = m and t[1] > 0 then r:= t[2]
         fi
       od;
       r
 end proc:
map(f, [$1..100]); # Robert Israel, Jul 21 2025

apply( {A384103(n)=for(x=1,n\/5, my(p=6*x+1, q=6*x-1, y=if((n-x)%p==0, (n-x)\p, (n+x)%p==0, -(n+x)\p, (n-x)%q==0, (n-x)\q, (n+x)%q==0,-(n+x)\q)); y && abs(y) <= x && return(y))}, [1..90])

A120382 Even numbers k such that 3k-1 and 3k+1 are not prime.

Original entry on oeis.org

40, 48, 62, 68, 72, 82, 96, 100, 108, 114, 138, 142, 158, 172, 176, 178, 184, 194, 208, 212, 222, 232, 238, 260, 264, 268, 272, 278, 282, 290, 298, 300, 308, 320, 334, 342, 348, 352, 358, 360, 378, 380, 382, 386, 392, 402, 414, 418, 422, 424, 438, 444, 446, 448, 450

Offset: 1

Pierre CAMI, Jun 29 2006

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

Cf. A060461.

remove(k -> isprime(3*k-1) or isprime(3*k+1), [seq(k,k=2..1000,2)]); # Robert Israel, Sep 30 2024

Select[Range[2,450,2],!PrimeQ[3#-1]&&!PrimeQ[3#+1]&] (* James C. McMahon, Sep 30 2024 *)
Select[2*Range[300],NoneTrue[3#+{1,-1},PrimeQ]&] (* Harvey P. Dale, May 27 2025 *)

isok(n) = !(n%2) && !isprime(3*n-1) && !isprime(3*n+1); \\ Michel Marcus, Sep 15 2019

a(n) = 2*A060461(n). - Michel Marcus, Sep 15 2019

More terms from Michel Marcus, Sep 15 2019

cp's OEIS Frontend

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

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A171696 Nonnegative numbers k such that neither of 6*k +- 1 is prime.

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A121765 Numbers n such that 6*n-1 is composite while 6*n+1 is prime.

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A121763 Numbers n such that 6*n-1 is prime while 6*n+1 is composite.

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A171697 1 together with pairs of composites of the form (6n-1, 6n+1).

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A157966 Number of 3's in A157733(n).

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A259826 Numbers n such that n is a multiple of 6 and both n-1 and n+1 are composite.

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A121765 Numbers n such that 6n-1 is composite while 6n+1 is prime.

A121763 Numbers n such that 6n-1 is prime while 6n+1 is composite.

A120382 Even numbers k such that 3k-1 and 3k+1 are not prime.