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A350542
Twin primes x, represented by their average, such that x is the first and x+30 the last of four successive twins.
2
12, 626598, 663570, 1322148, 2144478, 2668218, 6510192, 6937938, 10187910, 11495580, 11721768, 18873498, 18873510, 25658430, 39659532, 39851292, 46533468, 80572158, 84099318, 86944602, 91814712, 93956100, 123911532, 128469150, 129902022, 148979838
OFFSET
1,1
COMMENTS
Subsequence of A014574. For x>6, d=30 is the least possible difference between the least and the greatest of four twins. With d=24, six primes would have the form 6*k+-1, 6*k+6+-1,6*k+12+-1 which is impossible because one of the six numbers would be divisible by 5. Therefore, d>24, except for x=6. The distribution of 1134 terms < 10^11 is in accordance with the k-tuple conjecture, see links "k-tuple conjecture" and A350541, "Test of the k-tuple conjecture".
Generalization:
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 A350541
4 24 Only one quintuple: (6,12,18,30,42)
4 30 Current sequence
5 36 See A350543
5 42 See A350543
5 48 A350543
LINKS
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.
EXAMPLE
Quadruples of twins Example 8-tuple of primes
(x,x+ 6,x+18,x+30) x=12 (11,13,17,19,29,31,41,43)
(x,x+12,x+24,x+30) x=626598 (x-1,x+1,x+11,x+13,x+23,x+25,x+29,x+31)
(x,x+12,x+18,x+30) x=663570 (x-1,x+1,x+11,x+13,x+17,x+19,x+29,x+31)
(x,x+ 6,x+24,x+30), (x,x+6,x+12,x+30) and (x,x+18,x+24,x+30) do not occur for divisibility reasons.
MATHEMATICA
Select[Prime@Range[4, 200000], Count[Range[#, #+30], _?(PrimeQ@#&&PrimeQ[#+2]&)]==4&]+1 (* Giorgos Kalogeropoulos, Jan 07 2022 *)
PROG
(Maxima)
block(twin:[], n:0, p1:11, j2:1, nmax: 3,
/*returns nmax terms*/
m:4, d:30, 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 07 2022
STATUS
approved