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

Author

Gerhard Kirchner, Jan 07 2022

Keywords

• nonn

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.

Examples

			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
		

Links

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

Crossrefs

Cf. A014574, A173037, A350541, A350542.

Programs

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