A350541 -id:A350541 - cp's OEIS frontend

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

Gerhard Kirchner, Jan 07 2022

nonn

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

			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.

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

Cf. A014574, A173037, A350541, A350543.

Select[Prime@Range[4,200000], Count[Range[#,#+30],?(PrimeQ@#&&PrimeQ[#+2]&)]==4&]+1 (* _Giorgos Kalogeropoulos, Jan 07 2022 *)

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

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

Original entry on oeis.org

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

Offset: 1

Gerhard Kirchner, Jan 07 2022

nonn

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.

			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

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

Cf. A014574, A173037, A350541, A350542.

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

cp's OEIS Frontend

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

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