A350541 Twin primes x, represented by their average, such that x is the first and x+18 the last of three successive twins.

12, 180, 810, 5640, 9420, 18042, 62970, 88800, 97842, 109830, 165702, 284730, 392262, 452520, 626610, 663570, 663582, 855720, 983430, 1002342, 1003350, 1068702, 1146780, 1155612, 1322160, 1329702, 1592862, 1678752, 1718862, 1748472, 2116560, 2144490

Offset: 1

Author

Gerhard Kirchner, Jan 06 2022

Keywords

• nonn

Comments

Subsequence of A014574. For x>6, d=18 is the least possible difference between the least and the greatest of three twins. For d=12, one of the six terms 6*k+-1, 6*k+6+-1,6*k+12+-1 would be divisible by 5. Therefore, d>12, except for x=6.

The distribution of 35314 terms < 10^11 is in accordance with the k-tuple conjecture, see links "k-tuple conjecture" and "Test of the k-tuple conjecture".

Generalizations:

Twin primes x such that x is the first and x+d the last of m successive twins.

m d

1 0 A014574(n) twin primes

2 6 A173037(n)-3

3 12 Only one quadruple: (6,12,18,30)

3 18 Current sequence

4 24 Only one quintuple: (6,12,18,30,42)

4 30 A350542

5 36 See A350543

5 42 See A350543

5 48 A350543

Examples

			Triples of twins Example   6-tuple of primes
(x,x+ 6,x+18)     x= 12   (11,13,17,19,29,31)
(x,x+12,x+18)     x=180   (179,181,191,193,197,199)
		

Links

• Gerhard Kirchner, Test of the k-tuple conjecture
• Eric Weisstein's World of Mathematics, k-tuple conjecture.

Crossrefs

Cf. A014574, A173037, A350542, A350543.

Programs

Mathematica

Select[Prime@Range[4,160000],Count[Range[#,#+18],?(PrimeQ@#&&PrimeQ[#+2]&)]==3&]+1 (* _Giorgos Kalogeropoulos, Jan 07 2022 *)

Maxima

block(twin:[], n:0, p1:11, j2:1, nmax: 3,
/*returns nmax terms*/
m:3, d:18, w: makelist(-d,i,1,m),
while n