A350542 Twin primes x, represented by their average, such that x is the first and x+30 the last of four successive twins.

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

Table of n, a(n) for n=1..26.

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

Cf. A014574, A173037, A350541, A350543.

Sequence in context: A055312 A365242 A296138 * A159951 A283627 A013862

Adjacent sequences: A350539 A350540 A350541 * A350543 A350544 A350545

Keyword

nonn

Author

Gerhard Kirchner, Jan 07 2022

Status

approved