A173037 -id:A173037 - cp's OEIS frontend

A350541 Twin primes x, represented by their average, such that x is the first and x+18 the last of three successive twins.

12, 180, 810, 5640, 9420, 18042, 62970, 88800, 97842, 109830, 165702, 284730, 392262, 452520, 626610, 663570, 663582, 855720, 983430, 1002342, 1003350, 1068702, 1146780, 1155612, 1322160, 1329702, 1592862, 1678752, 1718862, 1748472, 2116560, 2144490

Offset: 1

Gerhard Kirchner, Jan 06 2022

nonn

Subsequence of A014574. For x>6, d=18 is the least possible difference between the least and the greatest of three twins. For d=12, one of the six terms 6*k+-1, 6*k+6+-1,6*k+12+-1 would be divisible by 5. Therefore, d>12, except for x=6.
The distribution of 35314 terms < 10^11 is in accordance with the k-tuple conjecture, see links "k-tuple conjecture" and "Test of the k-tuple conjecture".
Generalizations:
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  Current sequence
4  24  Only one quintuple: (6,12,18,30,42)
4  30  A350542
5  36  See  A350543
5  42  See  A350543
5  48  A350543

			Triples of twins Example   6-tuple of primes
(x,x+ 6,x+18)     x= 12   (11,13,17,19,29,31)
(x,x+12,x+18)     x=180   (179,181,191,193,197,199)

Gerhard Kirchner, Test of the k-tuple conjecture
Eric Weisstein's World of Mathematics, k-tuple conjecture.

Cf. A014574, A173037, A350542, A350543.

Select[Prime@Range[4,160000],Count[Range[#,#+18],?(PrimeQ@#&&PrimeQ[#+2]&)]==3&]+1 (* _Giorgos Kalogeropoulos, Jan 07 2022 *)

block(twin:[], n:0, p1:11, j2:1, nmax: 3,
/*returns nmax terms*/
m:3, d:18, w: makelist(-d,i,1,m),
while n

A173092 Numbers k such that 3k-4, 3k-2, 3k+2, and 3k+4 are primes.

Original entry on oeis.org

3, 5, 35, 65, 275, 495, 625, 695, 1085, 1155, 1885, 3145, 4335, 5215, 5245, 5355, 6015, 6305, 6475, 7005, 7425, 8435, 10575, 11615, 14595, 17115, 18445, 20995, 22405, 23165, 24075, 25755, 26565, 27015, 27575, 29605, 32615, 33045, 33705, 36615, 38845, 39765, 40735, 45155, 48055, 52425

Offset: 1

Juri-Stepan Gerasimov, Feb 10 2010, Feb 19 2010

			3 is a term because 3*3-4=5, 3*3-2=7, 3*3+2=11, and 3*3+4=13 are all prime.

Cf. A173037.

[ n: n in [0..60000] | IsPrime(3*n-2) and IsPrime(3*n+2) and IsPrime(3*n-4) and IsPrime(3*n+4) ]; // Vincenzo Librandi, Dec 04 2010

Select[Range[10^5], PrimeQ[3# - 4]&&PrimeQ[3# - 2] && PrimeQ[3# + 2] && PrimeQ[3# + 4]&] (* Alonso del Arte, Dec 04 2010 *)

a(n) = A173037(n)/3.

Entries checked by D. S. McNeil, Nov 26 2010

Extended by Vincenzo Librandi and Charles R Greathouse IV, Mar 25 2010

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

Original entry on oeis.org

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

A178082 Numbers k for which 5k-4, 5k-2, 5k+2, and 5k+4 are primes.

Original entry on oeis.org

3, 21, 39, 165, 297, 375, 417, 651, 693, 1131, 1887, 2601, 3129, 3147, 3213, 3609, 3783, 3885, 4203, 4455, 5061, 6345, 6969, 8757, 10269, 11067, 12597, 13443, 13899, 14445, 15453, 15939, 16209, 16545, 17763, 19569, 19827, 20223, 21969, 23307

Offset: 1

Roger L. Bagula, May 19 2010

			The associated prime quadruplets start as:
     11,    13,    17,    19;   (for n =  3)
    101,   103,   107,   109;   (for n = 21)
    191,   193,   197,   199;   (for n = 39)
    821,   823,   827,   829;
   1481,  1483,  1487,  1489;
   1871,  1873,  1877,  1879;
   2081,  2083,  2087,  2089;
   3251,  3253,  3257,  3259;
   3461,  3463,  3467,  3469;
   5651,  5653,  5657,  5659;
   9431,  9433,  9437,  9439;
  13001, 13003, 13007, 13009;
  15641, 15643, 15647, 15649;
  15731, 15733, 15737, 15739;
  16061, 16063, 16067, 16069;
  18041, 18043, 18047, 18049;
  18911, 18913, 18917, 18919;
  19421, 19423, 19427, 19429.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007811, A024897, A024896.

[n: n in [0..1000]| IsPrime(5*n - 4) and IsPrime(5*n - 2) and IsPrime(5*n + 2) and IsPrime(5*n + 4)]; // Vincenzo Librandi, Nov 30 2010

Flatten[Table[If[PrimeQ[5*n + 2] && PrimeQ[5*n - 2] && PrimeQ[5*n + 4] && PrimeQ[5*n - 4], n, {}], {n, 0, 10000}]]
Select[Range[25000],AllTrue[5#+{4,2,-2,-4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 03 2018 *)

a(n) = A173037(n+1)/5.

A173135 Primes other than 3 and 5.

Original entry on oeis.org

2, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Juri-Stepan Gerasimov, Feb 10 2010

Cf. A173037, A173092.

a(n)=if(n>1,prime(n+2),2) \\ Charles R Greathouse IV, Aug 26 2011

Better definition from Charles R Greathouse IV, Mar 24 2010, corrected Apr 22 2010

A176002 Numbers n such that 15*prime(n)+{-4,-2,2,4} are all primes.

Original entry on oeis.org

4, 6, 34, 176, 608, 1023, 1338, 1377, 1555, 1980, 2054, 2850, 2893, 3061, 3263, 3572, 3977, 4029, 4244, 4405, 6099, 6548, 7203, 7348, 7350, 7572, 7574, 9028, 10657, 11976, 12215, 12874, 13247, 13388, 13432, 14537, 14813, 15115, 15412, 15509

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2010

nonn

Numbers n such that 15*prime(n)-4, 15*prime(n)-2, 15*prime(n)+2 and 15*prime(n)+4 are primes.

			a(1)=4 because 15*prime(4)-4=101, 15*prime(4)-2=103, 15*prime(4)+2=107 and 15*prime(4)+4=109.

Cf. A112540, A173037, A173092.

p15Q[n_]:=And@@PrimeQ/@(15 Prime[n]+{-4,-2,2,4}); Select[Range[16000], p15Q]  (* Harvey P. Dale, Mar 20 2011 *)

A000040(a(n))=A112540(k).

More terms from R. J. Mathar, Apr 16 2010

cp's OEIS Frontend

A350541 Twin primes x, represented by their average, such that x is the first and x+18 the last of three successive twins.

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A173092 Numbers k such that 3k-4, 3k-2, 3k+2, and 3k+4 are primes.

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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.

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

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A178082 Numbers k for which 5*k-4, 5*k-2, 5*k+2, and 5*k+4 are primes.

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A173135 Primes other than 3 and 5.

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A176002 Numbers n such that 15*prime(n)+{-4,-2,2,4} are all primes.

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Extensions

A178082 Numbers k for which 5k-4, 5k-2, 5k+2, and 5k+4 are primes.