A067611 -id:A067611 - cp's OEIS frontend

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.

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])

A173229 a(n) is the n-th number m such that 6m-1 is composite minus the n-th number k such that 6k+1 is composite.

Original entry on oeis.org

2, 3, 4, 2, 5, 2, 4, 4, 3, 3, 5, 4, 2, 1, 2, 5, 6, 7, 7, 6, 6, 6, 4, 7, 5, 4, 6, 4, 4, 4, 4, 5, 5, 6, 5, 7, 8, 7, 6, 6, 6, 8, 8, 9, 9, 10, 8, 8, 12, 8, 9, 8, 9, 8, 8, 8, 7, 8, 7, 8, 6, 4, 3, 4, 4, 6, 7, 6, 6, 6, 8, 6, 6, 5, 5, 6, 8, 7, 10, 9, 9, 9, 11, 11, 11, 12, 11, 10, 9, 7, 10, 8, 8, 6, 6, 6, 4, 5, 5, 7

Offset: 1

Juri-Stepan Gerasimov, Feb 13 2010, Feb 14 2010

nonn

A046953 U A046954(without zero) = A067611 where A067611 U A002822 U A171696 = A001477.

			a(1) = 6 - 4 = 2;
a(2) = 11 - 8 = 3;
a(3) = 13 - 9 = 4.

Cf. A001477, A002822, A067611, A046953, A046954, A171696.

A046953 := proc(n) if n = 1 then 6 ; else for a from procname(n-1)+1 do if not isprime(6*a-1) then return a; end if; end do: end if; end proc:
A046954 := proc(n) if n = 1 then 0 ; else for a from procname(n-1)+1 do if not isprime(6*a+1) then return a; end if; end do: end if; end proc:
A173229 := proc(n) A046953(n)-A046954(n+1) ; end proc:
seq(A173229(n),n=1..120) ; # R. J. Mathar, May 02 2010

a(n) = A046953(n) - A046954(n+1).

Corrected from a(63) onwards by R. J. Mathar, May 02 2010

A173231 a(n) is the n-th number m such that 6m-1 is composite plus the n-th number k such that 6k+1 is composite.

Original entry on oeis.org

10, 19, 22, 30, 35, 40, 44, 48, 51, 59, 63, 66, 70, 73, 80, 87, 90, 93, 95, 102, 104, 106, 110, 115, 119, 122, 126, 132, 134, 138, 142, 147, 153, 156, 161, 165, 168, 171, 174, 176, 178, 184, 186, 193, 195, 198, 202, 204, 210, 216, 221, 224, 227, 230, 234, 236

Offset: 1

Juri-Stepan Gerasimov, Feb 13 2010, Feb 15 2010

nonn

A001477 = A002822 U A171696 U A067611 where A067611 = A121763 U A121765; A121763 = A046954 U A171696 and A121765 = A046953 U A171696.

			a(1) = 6 + 4 = 10;
a(2) = 11 + 8 = 19;
a(3) = 13 + 9 = 22.

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

Cf. A001477, A002822, A067611, A046953, A046954, A121763, A121765, A171696.

A046953:=Filtered([1..250], k-> not IsPrime(6*k-1));;
A046954:=Filtered([0..250], n-> not IsPrime(6*n+1));;
Print(List([1..80], j->A046953[j]+A046954[j+1])); # G. C. Greubel, Feb 21 2019

A046953:=[n: n in [1..250] | not IsPrime(6*n-1)];
A046954:=[n: n in [0..250] | not IsPrime(6*n+1)];
[A046953[n] + A046954[n+1]: n in [1..80]]; // G. C. Greubel, Feb 21 2019

A046953 := proc(n) if n = 1 then 6 ; else for a from procname(n-1)+1 do if not isprime(6*a-1) then return a; end if; end do: end if; end proc:
A046954 := proc(n) if n = 1 then 0 ; else for a from procname(n-1)+1 do if not isprime(6*a+1) then return a; end if; end do: end if; end proc:
A173231 := proc(n) A046953(n)+A046954(n+1) ; end proc:
seq(A173231(n),n=1..120) ; # R. J. Mathar, May 02 2010

A046953:= Select[Range[250], !PrimeQ[6#-1] &];
A046954:= Select[Range[0, 250], !PrimeQ[6#+1] &];
Table[A046953[[n]] +A046954[[n+1]], {n,1,80}]

A046953=[n for n in (1..250) if not is_prime(6*n-1)];
A046954=[n for n in (0..250) if not is_prime(6*n+1)];
[A046953[n] + A046954[n+1] for n in (0..80)] # G. C. Greubel, Feb 21 2019

a(n) = A046953(n) + A046954(n+1).

Entries checked by R. J. Mathar, May 02 2010

A304978 Numbers that can be expressed in more than one way as 6xy + x + y with x >= y > 0.

Original entry on oeis.org

106, 155, 197, 204, 253, 288, 302, 351, 379, 400, 421, 449, 470, 498, 504, 535, 547, 554, 561, 596, 645, 652, 687, 694, 704, 729, 743, 779, 782, 792, 820, 834, 841, 873, 890, 904, 925, 939, 953, 988, 1016, 1029, 1037, 1042, 1054, 1079, 1086, 1107, 1121, 1135, 1184, 1198, 1204, 1211, 1219, 1233, 1254, 1276, 1282, 1289, 1329

Offset: 1

Pedro Caceres, May 22 2018

nonn

Is it possible to find a closed form formula for this sequence?

Numbers k such that 6*k+1 has at least 5 divisors == 1 (mod 6). - Robert Israel, Jan 20 2019

			106 is in this sequence because 106 can be expressed in two different ways as 6xy + x + y: 6*8*2 + 8 + 2 and 6*15*1 + 15 + 1.

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

Subsequence of A067611. A279060.

filter:= proc(n) nops(select(t -> t mod 6 =1, numtheory:-divisors(6*n+1)))>= 5 end proc:
select(filter, [$1..2000]); # Robert Israel, Jan 20 2019

Select[Range[1329], 2 == Length@ FindInstance[ 6*x*y+x+y == # && x >= y > 0, {x, y}, Integers, 2] &] (* Giovanni Resta, May 29 2018 *)

is(n) = my(i=0); for(x=1, n, for(y=1, x, if(n==6*x*y+x+y, i++; if(i==2, return(1))))); 0 \\ Felix Fröhlich, May 29 2018

from sympy import divisors
def ok(n): return sum(d%6 == 1 for d in divisors(6*n+1)) > 4
print([n for n in range(1330) if ok(n)]) # David Radcliffe, Jun 19 2025

A054775 Positive multiples of 6 which are not the midpoint of a pair of twin primes.

Original entry on oeis.org

24, 36, 48, 54, 66, 78, 84, 90, 96, 114, 120, 126, 132, 144, 156, 162, 168, 174, 186, 204, 210, 216, 222, 234, 246, 252, 258, 264, 276, 288, 294, 300, 306, 318, 324, 330, 336, 342, 354, 360, 366, 372, 378, 384, 390, 396, 402, 408, 414, 426, 438, 444, 450, 456, 468, 474

Offset: 1

Stuart M. Ellerstein (ellerstein(AT)aol.com), May 19 2000

nonn

Cf. A067611 (sequence divided by 6).

[ n: n in [6..600 by 6] | not IsPrime(n-1) or not IsPrime(n+1) ];  // Klaus Brockhaus, Aug 18 2010

forstep(n=6,600,6,if(!(isprime(n-1)&&isprime(n+1)),print1(n,","))) \\ Klaus Brockhaus, Aug 18 2010

Corrected by Zak Seidov and Klaus Brockhaus, Aug 18 2010

Edited by N. J. A. Sloane, Aug 19 2010

cp's OEIS Frontend

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.

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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.

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A173229 a(n) is the n-th number m such that 6m-1 is composite minus the n-th number k such that 6k+1 is composite.

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A173231 a(n) is the n-th number m such that 6*m-1 is composite plus the n-th number k such that 6*k+1 is composite.

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A304978 Numbers that can be expressed in more than one way as 6xy + x + y with x >= y > 0.

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A054775 Positive multiples of 6 which are not the midpoint of a pair of twin primes.

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Extensions

A173231 a(n) is the n-th number m such that 6m-1 is composite plus the n-th number k such that 6k+1 is composite.