A350543 Twin primes x, represented by their average, such that x is the first and x+48 the last of five successive twins.

12, 3919212, 325267932, 905119332, 2013256362, 3066212112, 3240097962, 4046054430, 6567515370, 7561533402, 10816172202, 10895874132, 17444777880, 20905115040, 22194295812, 23641113912, 26079344100, 26368755222, 27350615220, 29861090682, 33240296052

Offset

1,1

Comments

Subsequence of A014574. The terms represent quintuples of twin primes. As there are only 31 terms < 10^11, the accordance with the k-tuple conjecture is not very good, see links "k-tuple conjecture" and A350541, "Test of the k-tuple conjecture". Moreover, the formalism of the conjecture allows the evaluation of the expected frequencies of eight types of quintuples relative to the frequency of all quintuples. The differences are considerable:

relative frequencies

Example observed expected

(1) 11/31=35.5% 23.7%

(2) 5/31=16.1% 15.0%

(3) 3/31= 9.7% 7.5%

(4) 6/31=19.4% 23.7%

(5) 3/31= 9.7% 15.0%

(6) 2/31= 6.5% 3.8%

(7) 1/31= 3.2% 7.5%

(8) 0 3.8%

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 A350542

5 36 6, 39713433660, 66419473020, 71525244600*

5 42 18873492, 180929682, 1170073332, 2550576612, 5807487204, 27523454232, 33497368554, 50062053714, 63167632254, 86883508944, 99939276954*

5 48 Current sequence

Annotations:

*The number of terms < 10^11 is too small for submitting a new sequence.

(m=5,d=30) is empty for divisibility reasons.

Links

Table of n, a(n) for n=1..21.

Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Example

The quintuples of twins have the form (x,x+a,x+b,x+c,x+d)
  (a,b,c,d)      least example
(1)  6,18,30,48  x=           12
(2)  6,30,36,48  x=    123919212
(3)  6,18,36,48  x= 123240097962
(4) 18,30,42,48  x= 124046054430
(5) 12,18,42,48  x=1217444777880
(6) 12,18,30,48  x=1220905115040
(7) 12,30,42,48  x=1227350615220
(8) 18,30,42,48  x>10^11

Prog

(Maxima)
block(twin:[6], n:1,  p1:11,  j2:1, nmax: 3,
/*returns nmax terms*/
m:5, d:48, 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, A350542.

Sequence in context: A283627 A013862 A116233 * A145745 A175906 A144546

Adjacent sequences: A350540 A350541 A350542 * A350544 A350545 A350546

Keyword

nonn

Author

Gerhard Kirchner, Jan 07 2022

Status

approved