A002822 -id:A002822 - cp's OEIS frontend

A176114 Numbers n such that 30n-1, 30n+1 are twin primes.

1, 2, 5, 6, 8, 9, 14, 19, 20, 22, 27, 34, 35, 41, 43, 44, 54, 65, 71, 77, 78, 85, 91, 93, 99, 100, 104, 110, 111, 112, 113, 118, 131, 134, 135, 141, 142, 155, 160, 167, 170, 176, 184, 188, 195, 196, 203, 209, 210, 212, 215, 219, 222, 223, 226, 229, 232, 245, 252

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 08 2010

nonn

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A002822.

Select[Range[6! ],PrimeQ[30*#-1]&&PrimeQ[30*#+1]&]
Select[Range[300],AllTrue[30#+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 21 2018 *)

A060210 Largest prime factor of 1+smaller term of twin primes.

Original entry on oeis.org

2, 3, 3, 3, 5, 7, 5, 3, 17, 3, 23, 5, 5, 3, 11, 19, 5, 5, 47, 13, 29, 7, 3, 11, 29, 19, 5, 103, 107, 11, 5, 137, 23, 13, 7, 17, 43, 7, 59, 13, 3, 41, 71, 43, 31, 11, 17, 11, 19, 31, 67, 5, 139, 283, 41, 149, 13, 313, 23, 13, 37, 13, 347, 29, 11, 71, 17, 373, 7, 11, 13, 397, 17

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Also: Largest prime factor of the average, or the sum, of twin prime pairs. - M. F. Hasler, Jan 03 2011

			101 is the 9th lesser twin, 102 = 2*3*17, and its max p factor is 17=a(9).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A001359, A001454, A002822, A059960, A027856.

FactorInteger[1+#][[-1,1]]&/@Select[Partition[Prime[Range[500]],2,1], #[[2]]- #[[1]]==2&][[All,1]] (* Harvey P. Dale, Jan 16 2017 *)

p=3; for(n=1,1e3, until(o+2==p,p=nextprime(2+o=p)); print1(vecmax(factor(p-1)[,1])","))  \\ M. F. Hasler, Jan 03 2011

A057767 Number of twin prime pairs between P(n)^2 and P(n+1)^2 where P(n) is the n-th prime.

Original entry on oeis.org

1, 2, 2, 4, 2, 7, 2, 4, 8, 2, 11, 7, 3, 11, 13, 13, 5, 19, 11, 3, 15, 14, 14, 21, 15, 7, 10, 6, 11, 42, 12, 27, 6, 45, 10, 20, 17, 21, 23, 25, 13, 49, 7, 20, 8, 52, 59, 23, 9, 16, 32, 9, 46, 33, 27, 43, 7, 30, 20, 12, 68, 88, 22, 18, 24, 88, 41, 70, 14

Offset: 1

Naohiro Nomoto, Oct 31 2000

nonn

Conjecture: this sequence is always positive.
For n > 1 also the number of twin ranks k in A002822 between M(n) and M(n+1), where M(n) = (P(n)^2-1)/6. (Indeed, none of the three numbers {6k-1, 6k, 6k+1} will ever be equal to P(n)^2 if 6k+-1 are twin primes, therefore P(n)^2 <= 6k-1 < 6k+1 <= P(n+1)^2 <=> (P(n)^2-1)/6 <= k <= (P(n+1)^2-1)/6.) The twin prime conjecture is equivalent to say a(n) > 0 for infinitely many n. - M. F. Hasler, Jun 26 2019
Records of "lows" (such that a(k) > a(m) for all k > m) are (conjectured): a(1) = 1, a(10) = 2, a(20) = 3, a(33) = 6, a(57) = 7, a(89) = 10, a(140) = 19, a(190) = 21, a(236) = 30, a(256) = 33, a(265) = 35, a(307) = 42, a(346) = 43, a(384) = 44, a(495) = 51, a(498) = 55, a(545) = 62, a(555) = 68, a(613) = 71, a(643) = 76, a(673) = 79, a(719) = 87, a(723) = 93, a(755) = 94, a(772) = 96, a(872) = 98, a(936) = 107, ... None of these is proven, each one would imply the twin prime conjecture. - M. F. Hasler, Jun 26 2019
Record lows of a(n)/n are: 1/1 = 2/2 = 1.0, 2/3 = 0.66667, 2/5 = 0.4, 2/7 = 0.28571, 2/10 = 0.2, 3/20 = 0.15, 7/57 = 0.12281, 10/89 = 0.11236, 21/190 = 0.11053, 51/495 = 0.10303, 342/3435 = 0.099563, 716/7202 = 0.099417, 797/8126 = 0.098080, 793/8155 = 0.097241, 817/8463 = 0.096538, 892/9406 = 0.094833, ... - M. F. Hasler, Jun 27 2019

			From _M. F. Hasler_, Jun 26 2019: (Start)
Between P(1)^2 = 2^2 = 4 and P(2)^2 = 3^2 = 9 there is only the twin prime pair (5,7), whence a(1) = 1.
Between P(2)^2 = 3^2 = 9 and P(3)^2 = 5^2 = 25 there are the twin prime pairs (11,13) and (17,19) whence a(2) = 2.
Between P(3)^2 = 5^2 = 25 and P(4)^2 = 7^2 = 49 there are the twin prime pairs (29,31) and (41,43) whence a(3) = 2.
Between P(4)^2 = 7^2 = 49 and P(5)^2 = 11^2 = 121 there are the twin prime pairs (59,61), (71,73), (101,103) and (107,109), whence a(4) = 4.
etc. (End)

M. F. Hasler, Table of n, a(n) for n = 1..10000
A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

cp[{x_,y_}]:=Count[Partition[Range[x+1,y-1],3,1],?(AllTrue[{#[[1]],#[[3]]},PrimeQ]&)]; cp/@ Partition[Prime[Range[100]]^2,2,1] (* _Harvey P. Dale, Jul 27 2024 *)

A057767(n,c=0)={forprime(q=2+p=nextprime(prime(n)^2),prime(n+1)^2,p+2==(p=q)&&c++); c} \\ (Replaces slower code from Jun 26 2019.) - M. F. Hasler, Jul 04 2019

Offset corrected to 1 by M. F. Hasler, Jun 26 2019

A120875 Product of twin primes minus 1.

Original entry on oeis.org

14, 34, 142, 322, 898, 1762, 3598, 5182, 10402, 11662, 19042, 22498, 32398, 36862, 39202, 51982, 57598, 72898, 79522, 97342, 121102, 176398, 186622, 213442, 272482, 324898, 359998, 381922, 412162, 435598, 656098, 675682, 685582, 736162

Offset: 1

Lekraj Beedassy, Jul 09 2006

nonn

This sequence is a subsequence of A023515.

Jason Kimberley (using T. Noe's b037074), Table of n, a(n) for n = 1..10000

Cf. A002822, A023515, A037074, A075369, A120876.

Times[#, # + 2] - 1 & /@ Select[Prime@ Range@ 150, PrimeQ[# + 2] &] (* Michael De Vlieger, Oct 23 2015 *)

for(n=1, 200, if(prime(n+1)-prime(n)==2, print1(prime(n)*prime(n+1)-1", "))) \\ Altug Alkan, Oct 23 2015

a(n) = A037074(n)-1 = (A014574(n))^2 -2 = A075369(n)-2.
a(n) = 2*A120876(n). - Jason Kimberley, Oct 23 2015
a(n) = 36*A002822(n-1)^2-2, for n>1. - Jason Kimberley, Oct 23 2015
a(n) = A023515(A107770(n)). - Jason Kimberley, Oct 23 2015

A120876 (Product of twin primes - 1)/2.

Original entry on oeis.org

7, 17, 71, 161, 449, 881, 1799, 2591, 5201, 5831, 9521, 11249, 16199, 18431, 19601, 25991, 28799, 36449, 39761, 48671, 60551, 88199, 93311, 106721, 136241, 162449, 179999, 190961, 206081, 217799, 328049, 337841, 342791, 368081, 388961, 520199, 532511, 551249, 563921

Offset: 1

Lekraj Beedassy, Jul 09 2006

nonn

This sequence is a subsequence of A102770.

Jason Kimberley, Table of n, a(n) for n = 1..5000

Cf. The subsequence A086870.

(Times@@#-1)/2&/@Select[Partition[Prime[Range[200]], 2,1],Last[#]- First[#]== 2&] (* Harvey P. Dale, Jun 26 2011 *)

for(n=1, 200, if(prime(n+1)-prime(n)==2, print1((prime(n)*prime(n+1)-1)/2", "))) \\ Altug Alkan, Oct 23 2015

p=2; forprime(q=3, 1e3, if(q-p==2, print1(p*q\2", ")); p=q) \\ Charles R Greathouse IV, Apr 01 2016

a(n) = A120875(n)/2 = A075369(n)/2-1 = A075369(n)^2/2-1.
a(n) = 18*A002822(n-1)^2-1, for n>1.
a(n) = A102770(A107770(n)). - Jason Kimberley, Nov 10 2015

Corrected by T. D. Noe, Oct 25 2006

Edited by Jason Kimberley, Oct 23 2015

A124520 Numbers k such that 14k - 1 and 14k + 1 are twin primes.

Original entry on oeis.org

3, 30, 33, 63, 75, 78, 93, 102, 123, 138, 153, 162, 165, 192, 195, 240, 252, 273, 297, 303, 342, 387, 393, 420, 435, 438, 450, 468, 483, 522, 525, 540, 588, 630, 633, 660, 663, 717, 738, 747, 750, 765, 798, 825, 837, 855, 957, 978, 993, 1023, 1032, 1062

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

			3 is in the sequence since 14*3 - 1 = 41 and 14*3 + 1 = 43 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A040040, A045753, A002822, A124065, A124518, A124519, A124521, A124522, A063983.

Select[Range[1100], And @@ PrimeQ[{-1, 1} + 14# ] &] (* Ray Chandler, Nov 16 2006 *)

Extended by Ray Chandler, Nov 16 2006

A124521 Numbers k such that 16k - 1 and 16k + 1 are twin primes.

Original entry on oeis.org

12, 15, 27, 72, 93, 117, 132, 162, 168, 195, 198, 210, 258, 267, 300, 327, 330, 345, 435, 468, 642, 765, 813, 855, 903, 912, 960, 978, 993, 1128, 1143, 1182, 1290, 1350, 1353, 1365, 1392, 1398, 1440, 1632, 1680, 1713, 1737, 1797, 1848, 1860, 1947, 1953, 1962

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

			12 is in the sequence since 16*12 - 1 = 191 and 16*12 + 1 = 193 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A040040, A045753, A002822, A124065, A124518, A124519, A124520, A124522, A063983.

A124521:=n->`if`(isprime(16*n-1) and isprime(16*n+1), n, NULL): seq(A124521(n), n=1..2000); # Wesley Ivan Hurt, Oct 10 2014

Select[Range[2000], And @@ PrimeQ[{-1, 1} + 16# ] &] (* Ray Chandler, Nov 16 2006 *)

Extended by Ray Chandler, Nov 16 2006

A167379 Let p and q be twin primes, excluding the pair (3,5). Then p+q is always divisible by 6 and we set a(n) = (p+q)/6.

Original entry on oeis.org

2, 4, 6, 10, 14, 20, 24, 34, 36, 46, 50, 60, 64, 66, 76, 80, 90, 94, 104, 116, 140, 144, 154, 174, 190, 200, 206, 214, 220, 270, 274, 276, 286, 294, 340, 344, 350, 354, 364, 384, 410, 426, 430, 434, 440, 476, 484, 494, 496, 536, 540, 556, 566, 574, 596, 624, 626

Offset: 1

Tanin (Mirza Sabbir Hossain Beg) (mirzasabbirhossainbeg(AT)yahoo.com), Nov 02 2009

nonn

By definition, q = p+2. Hence (p+q)/6 = (p+p+2)/6 = (2p+2)/6 = (p+1)/3. Thus a(n) = (1+A001359(n+1))/3. - Jonathan Vos Post, Nov 03 2009

			First (lesser of twin prime pair) excluding (3,5) = 5; (5+1)/3 = 2, hence A167379(1) = 2. The 10th (lesser of twin prime pair) excluding (3,5) = 137; (137+1)/3 = 46, hence A167379(10)= 46. - _Jonathan Vos Post_, Nov 03 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A002822. [Zak Seidov, Nov 02 2009]

[2*n: n in [1..630] | IsPrime(6*n+1) and IsPrime(6*n-1)]; // Vincenzo Librandi, Jun 13 2016

Total[#]/6&/@Select[Partition[Prime[Range[3,500]],2,1],#[[2]]-#[[1]] == 2&] (* Harvey P. Dale, Jan 30 2013 *)
2 Select[Range[35000], PrimeQ[6 # - 1] && PrimeQ[6 # + 1] &] (* Vincenzo Librandi, Jun 13 2016 *)

a(n) = 2*A002822(n). - R. J. Mathar, Nov 09 2009

a(n) = (1+A001359(n+1))/3. - Jonathan Vos Post, Nov 03 2009

Edited (but not checked) by N. J. A. Sloane, Nov 02 2009

Extended by R. J. Mathar, Nov 09 2009

A182482 6nA182481(n)-1.

Original entry on oeis.org

5, 11, 17, 71, 29, 71, 41, 191, 107, 59, 197, 71, 311, 419, 179, 191, 101, 107, 227, 239, 881, 659, 137, 431, 149, 311, 809, 2687, 347, 179, 1301, 191, 197, 1019, 419, 431, 1997, 227, 1871, 239, 1229, 2267, 1031, 1319, 269, 827, 281, 1151, 881, 599

Offset: 1

Vladimir Shevelev, May 01 2012

nonn

By the construction of A182481, every term is lesser of twin primes.
Every lesser more than 3 of twin primes appears in the sequence.
Number m=(a(n)+1)/6 is the place of the last appearance of a(n); m is multiple of all previous places of the appearance of a(n), if they exist.
In particular, a(n) appears only once, if (a(n)+1)/6 is 1 or prime (in this case n is 1 or prime and A182481(n)=1). Conversely is not true. For example, a(10)=59 appears only once, although 10 is not prime.

			All places where 71 appears are 4,6,12. "Thus" 12 is multiple of 4 and 6.
Since (101+1)/6=17 is prime, then 101 appears only once.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A182481, A001359, A006512, A014574, A001097, A077800, A002822.

A191626 Integers n such that both 6n and 36n fall between pairs of twin primes, that is, 6n-1, 6n+1, 36n-1, and 36n+1 are prime.

Original entry on oeis.org

2, 3, 5, 12, 23, 32, 45, 52, 58, 72, 107, 137, 138, 175, 182, 270, 278, 287, 325, 562, 577, 578, 597, 703, 747, 753, 872, 980, 1022, 1160, 1325, 1372, 1438, 1477, 1540, 1892, 1950, 2007, 2018, 2313, 2335, 2387, 2597, 2608, 2705, 2742, 2782, 3008

Offset: 1

Andrea Raffetti, Jul 11 2011

nonn

Infinite under Dickson's conjecture. [Charles R Greathouse IV, Jul 24 2011]

			5 is in the list because 5*6=30, 5*36=180 and both fall between a pair of twin primes (29,31 and 179,181).

Andrea Raffetti, Table of n, a(n) for n = 1..999

Subsequence of A002822.

Cf. A014574.

Select[Range[3100],And@@PrimeQ[{6#+1,6#-1,36#+1,36#-1}]&] (* Harvey P. Dale, Jul 27 2011 *)

cp's OEIS Frontend

A176114 Numbers n such that 30*n-1, 30*n+1 are twin primes.

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A060210 Largest prime factor of 1+smaller term of twin primes.

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A057767 Number of twin prime pairs between P(n)^2 and P(n+1)^2 where P(n) is the n-th prime.

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A120875 Product of twin primes minus 1.

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A120876 (Product of twin primes - 1)/2.

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A124520 Numbers k such that 14*k - 1 and 14*k + 1 are twin primes.

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A124521 Numbers k such that 16*k - 1 and 16*k + 1 are twin primes.

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A167379 Let p and q be twin primes, excluding the pair (3,5). Then p+q is always divisible by 6 and we set a(n) = (p+q)/6.

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A176114 Numbers n such that 30n-1, 30n+1 are twin primes.

A124520 Numbers k such that 14k - 1 and 14k + 1 are twin primes.

A124521 Numbers k such that 16k - 1 and 16k + 1 are twin primes.

A182482 6nA182481(n)-1.