OFFSET
1,1
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
LINKS
Gerhard Kirchner, Test of the k-tuple conjecture
Eric Weisstein's World of Mathematics, k-tuple conjecture.
EXAMPLE
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)
MATHEMATICA
Select[Prime@Range[4, 160000], Count[Range[#, #+18], _?(PrimeQ@#&&PrimeQ[#+2]&)]==3&]+1 (* Giorgos Kalogeropoulos, Jan 07 2022 *)
PROG
(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<nmax do(
p2: next_prime(p1), if p2-p1=2 then(
k:p1+1, j1:j2, j2:1+ mod(j2, m), w[j1]:k,
if w[j1]-w[j2]=d then(n:n+1, twin: append(twin, [k-d]))),
p1:p2), twin);
CROSSREFS
KEYWORD
nonn
AUTHOR
Gerhard Kirchner, Jan 06 2022
STATUS
approved