A002822 -id:A002822 - cp's OEIS frontend

A218829 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(m)) + 2 are both prime.

0, 0, 1, 2, 2, 3, 2, 3, 4, 2, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 3, 1, 3, 3, 1, 4, 4, 2, 3, 4, 4, 4, 4, 5, 3, 4, 4, 1, 4, 4, 3, 5, 4, 3, 3, 4, 6, 3, 5, 5, 3, 3, 3, 2, 4, 5, 4, 5, 4, 2, 3, 4, 4, 5, 5, 7, 4, 5, 2, 6, 4, 5, 7, 3, 5, 6, 2, 4, 3, 2

Offset: 1

Zhi-Wei Sun, Feb 05 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3, 22, 25, 38, 101, 273.
(ii) Each n = 2, 3, ... can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1 and prime(prime(m)) + 2 are all prime.
(iii) Any integer n > 5 can be written as k + m with k > 0 and m > 0 such that phi(k) - 1, phi(k) + 1 and prime(prime(m)) + 2 are all prime, where phi(.) is Euler's totient function.
(iv) If n > 2 is neither 10 nor 31, then n can be written as k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(prime(m))) + 2 are both prime.
(v) If n > 1 is not equal to 133, then n can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1 and prime(prime(prime(m))) + 2 are all prime.
Clearly, each part of the conjecture implies the twin prime conjecture.
We have verified part (i) for n up to 10^9. See the comments in A237348 for an extension of this part.

			a(3) = 1 since 3 = 2 + 1 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(1)) + 2 = prime(2) + 2 = 5 both prime.
a(22) = 1 since 22 = 20 + 2 with prime(20) + 2 = 71 + 2 = 73 and prime(prime(2)) + 2 = prime(3) + 2 = 5 + 2 = 7 both prime.
a(25) = 1 since 25 = 2 + 23 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(23)) + 2 = prime(83) + 2 = 431 + 2 = 433 both prime.
a(38) = 1 since 38 = 35 + 3 with prime(35) + 2 = 149 + 2 = 151 and prime(prime(3)) + 2 = prime(5) + 2 = 11 + 2 = 13 both prime.
a(101) = 1 since 101 = 98 + 3 with prime(98) + 2 = 521 + 2 = 523 and prime(prime(3)) + 2 = prime(5) + 2 = 11 + 2 = 13 both prime.
a(273) = 1 since 273 = 2 + 271 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(271)) + 2 = prime(1741) + 2 = 14867 + 2 = 14869 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Andrei-Lucian Dragoi, The "Vertical" Generalization of the Binary Goldbach's Conjecture as Applied on "Iterative" Primes with (Recursive) Prime Indexes (i-primeths), Journal of Advances in Mathematics and Computer Science (2017), Vol. 25, No. 2, pp. 1-32.
Zhi-Wei Sun, Unification of Goldbach's conjecture and the twin prime conjecture, a message to Number Theory List, Jan. 29, 2014.
Zhi-Wei Sun, Super Twin Prime Conjecture, a message to Number Theory List, Feb. 6, 2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000010, A000040, A001359, A002822, A006512, A072281, A182662, A199920, A236531, A236566, A236831, A236968, A237127, A237130, A237168, A237253, A237259, A237260, A237348.

pq[n_]:=PrimeQ[Prime[n]+2]
PQ[n_]:=PrimeQ[Prime[Prime[n]]+2]
a[n_]:=Sum[If[pq[k]&&PQ[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

A124518 Numbers k such that 10k-1 and 10k+1 are twin primes.

Original entry on oeis.org

3, 6, 15, 18, 24, 27, 42, 57, 60, 66, 81, 102, 105, 123, 129, 132, 162, 195, 213, 231, 234, 255, 273, 279, 297, 300, 312, 330, 333, 336, 339, 354, 393, 402, 405, 423, 426, 465, 480, 501, 510, 528, 552, 564, 585, 588, 609, 627, 630, 636, 645, 657, 666, 669, 678

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

All terms are divisible by 3. - Robert Israel, Apr 07 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A040040, A045753, A002822, A124065, A124519-A124522, A063983.

select(t -> isprime(10*t+1) and isprime(10*t-1), [seq(i,i=3..1000,3)]); # Robert Israel, Apr 07 2019

Select[Range[678], And @@ PrimeQ[{-1, 1} + 10# ] &] (* Ray Chandler, Nov 16 2006 *)

A236531 a(n) = |{0 < k < n: {6k -1 , 6k + 1} and {prime(n-k), prime(n-k) + 2} are both twin prime pairs}|.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 2, 3, 2, 3, 1, 4, 2, 3, 4, 1, 3, 2, 3, 5, 2, 4, 3, 2, 4, 1, 5, 4, 3, 5, 3, 3, 4, 3, 7, 5, 4, 7, 1, 7, 1, 5, 8, 3, 8, 5, 5, 5, 3, 9, 6, 6, 7, 4, 6, 3, 5, 8, 6, 7, 5, 6, 4, 5, 7, 7, 6, 5, 4, 4, 6, 5, 7, 6, 9, 3, 5, 5, 5, 6, 5, 8, 5, 5, 6, 5, 7, 4, 5, 10, 3, 7, 5, 6, 3, 4, 7, 5, 6, 6

Offset: 1

Zhi-Wei Sun, Jan 27 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 2.
(ii) If n > 3 is neither 11 nor 125, then n can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1, prime(m) + 2 and 3*prime(m) - 10 are all prime.
(iii) Any integer n > 458 can be written as p + q  with q > 0 such that {p, p + 2} and {prime(q), prime(q) + 2} are both twin prime pairs.
This is much stronger than the twin prime conjecture. We have verified part (i) of the conjecture for n up to 2*10^7.

			a(11) = 1 since {6*1 - 1, 6*1 + 1} = {5, 7} and {prime(10), prime(10) + 2} = {29, 31} are both twin prime pairs.
a(16) = 1 since {6*3 - 1, 6*3 + 1} = {17, 19} and {prime(13), prime(13) + 2} = {41, 43} are both twin prime pairs.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A001359, A002822, A006512, A199920.

p[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]
q[n_]:=PrimeQ[Prime[n]+2]
a[n_]:=Sum[If[p[k]&&q[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A060213 Lesser of twin primes whose average is 6 times a prime.

Original entry on oeis.org

11, 17, 29, 41, 101, 137, 281, 617, 641, 821, 1697, 1877, 2081, 2237, 2381, 2657, 2801, 3461, 3557, 3917, 4637, 4721, 5441, 6197, 6701, 8537, 8597, 9677, 10937, 12161, 12377, 12821, 12917, 13217, 13721, 13757, 13997, 14081, 16061, 17417

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Lowest factor-density among all positive consecutive integer triples; for p > 41, last digit of p can be only 1 or 7 (see Alexandrov link, p. 15). - Lubomir Alexandrov, Nov 25 2001

			102197 is here because 102198 = 17033*6 and 17033 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math/9811096 [math.NT], 1998, see p. 15.

Cf. A002822, A058383, A059960, A027856, A001454, A001359, A060212.

map(t -> 6*t-1, select(p -> isprime(p) and isprime(6*p-1) and isprime(6*p+1), [2,seq(i,i=3..10000,2)]));

Transpose[Select[Partition[Prime[Range[2500]],2,1],#[[2]]-#[[1]] == 2 && PrimeQ[Mean[#]/6]&]][[1]] (* Harvey P. Dale, May 04 2014 *)

isok(n) = isprime(n) && isprime(n+2) && !((n+1) % 6) && isprime((n+1)/6); \\ Michel Marcus, Dec 14 2013

a(n) = 6 * A060212(n) - 1. - Sean A. Irvine, Oct 31 2022

Offset changed to 1 by Michel Marcus, Dec 14 2013

A124065 Numbers k such that 8k - 1 and 8k + 1 are twin primes.

Original entry on oeis.org

9, 24, 30, 39, 54, 75, 129, 144, 165, 186, 201, 234, 261, 264, 324, 336, 339, 375, 390, 396, 420, 441, 459, 471, 516, 534, 600, 621, 654, 660, 690, 705, 735, 795, 819, 849, 870, 891, 936, 945, 1011, 1029, 1125, 1155, 1179, 1215, 1221, 1251, 1284, 1395, 1419

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

			9 is in the sequence since 8*9 - 1 = 71 and 8*9 + 1 = 73 are twin primes.

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

Intersection of A005122 and A005123.

Cf. A040040, A045753, A002822, A124518, A124519, A124520, A124521, A124522, A063983.

[n: n in [1..2000] | IsPrime(8*n+1) and IsPrime(8*n-1)] // Vincenzo Librandi, Mar 08 2010

Select[Range[1500], And @@ PrimeQ[{-1, 1} + 8# ] &] (* Ray Chandler, Nov 16 2006 *)

from sympy import isprime
def ok(n): return isprime(8*n - 1) and isprime(8*n + 1)
print(list(filter(ok, range(1420)))) # Michael S. Branicky, Sep 24 2021

Extended by Ray Chandler, Nov 16 2006

A121765 Numbers n such that 6n-1 is composite while 6n+1 is prime.

Original entry on oeis.org

6, 11, 13, 16, 21, 26, 27, 35, 37, 46, 51, 55, 56, 61, 62, 63, 66, 68, 73, 76, 81, 83, 90, 91, 96, 101, 102, 105, 112, 115, 118, 121, 122, 123, 125, 126, 128, 131, 142, 146, 151, 153, 156, 161, 165, 166, 168, 173, 178, 181, 186, 187, 188, 195, 200, 202, 206, 208, 216

Offset: 1

Lekraj Beedassy, Aug 20 2006

nonn

Entries in A046953 which are not in A060461 or equivalently, entries in A024899 which are not in A002822.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A121764.

Filtered([1..250], k-> not IsPrime(6*k-1) and IsPrime(6*k+1)); # G. C. Greubel, Feb 20 2019

[n: n in [1..250] | not IsPrime(6*n-1) and  IsPrime(6*n+1)]; // G. C. Greubel, Feb 20 2019

Select[Range[250], ! PrimeQ[6# - 1] && PrimeQ[6# + 1] &] (* Ray Chandler, Aug 22 2006 *)

for(n=1, 250, if(!isprime(6*n-1) && isprime(6*n+1), print1(n", "))) \\ G. C. Greubel, Feb 20 2019

[n for n in (1..250) if not is_prime(6*n-1) and  is_prime(6*n+1)] # G. C. Greubel, Feb 20 2019

Extended by Ray Chandler, Aug 22 2006

A182481 a(n) is the least k such that 6kn-1 and 6kn+1 are twin primes, and a(n)=0, if such k does not exist.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 4, 2, 1, 3, 1, 4, 5, 2, 2, 1, 1, 2, 2, 7, 5, 1, 3, 1, 2, 5, 16, 2, 1, 7, 1, 1, 5, 2, 2, 9, 1, 8, 1, 5, 9, 4, 5, 1, 3, 1, 4, 3, 2, 7, 1, 20, 5, 2, 8, 14, 1, 3, 21, 43, 4, 6, 3, 5, 8, 4, 9, 2, 1, 3, 1, 14, 15, 9, 30, 1, 4, 22, 7, 20, 21, 9

Offset: 1

Vladimir Shevelev, May 01 2012

nonn

Conjecture: a(n)>0; equivalently, for every n, the arithmetic progression {6*k*n-1} contains infinitely many lessers of twin primes (A001359).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

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

Table[k = 0; While[! (PrimeQ[6*k*n - 1] && PrimeQ[6*k*n + 1]), k++]; k, {n, 100}] (* T. D. Noe, May 02 2012 *)

a(n)=my(k);n*=6;until(isprime(n*k++-1)&&isprime(n*k+1),);k \\ Charles R Greathouse IV, May 01 2012

A329158 Let P1>=3, P2, P3 be consecutive primes, with P3-P2=2. a(n)=(P2+P3)/12 when P2-P1 sets a record.

Original entry on oeis.org

1, 2, 5, 25, 87, 192, 500, 1158, 1668, 4217, 4713, 5955, 17127, 28905, 61838, 76967, 96147, 139725, 260342, 1061923, 1205080, 4663498, 8871842, 11732765, 32534740, 42313103, 77638122, 92523718, 282054523, 728833340, 2940948542, 3344803093, 11810906035

Offset: 1

Hugo Pfoertner, Nov 06 2019

nonn

6*a(n)-1, 6*a(n)+1 are twin primes such that the prime gap immediately preceding 6*a(n)-1 sets a record. The corresponding gap lengths are provided in A329159.

Cf. A002822, A084105, A329159, A329160, A329161, A329164, A329165.

p1=3;p2=5;r=0;forprime(p3=7,1e9,if(p3-p2==2,d=p2-p1;if(d>r,r=d;print1((p2+p3)/12,", ")));p1=p2;p2=p3)

A329160 Let P1>=5, P2, P3 be consecutive primes, with P2-P1=2. a(n)=(P1+P2)/12 when P3-P2 sets a record.

Original entry on oeis.org

1, 5, 23, 33, 87, 278, 495, 1293, 2027, 2690, 4245, 6773, 13283, 24938, 28893, 44270, 67475, 139708, 224922, 315893, 971000, 1723960, 3319792, 6228255, 7013717, 13194622, 25321985, 31864375, 32163975, 65155398, 86090027, 381175405, 452803425

Offset: 1

Hugo Pfoertner, Nov 08 2019

nonn

6*a(n)-1, 6*a(n)+1 are twin primes such that the prime gap immediately following 6*a(n)+1 sets a record. The corresponding gap lengths are provided in A329161.

			See A329161.

Cf. A002822, A084105, A329158, A329159, A329161, A329164, A329165.

p1=5; p2=7; r=0; forprime(p3=11, 1e9, if(p2-p1==2, d=p3-p2; if(d>r, r=d; print1((p1+p2)/12, ", "))); p1=p2; p2=p3)

A121763 Numbers n such that 6n-1 is prime while 6n+1 is composite.

Original entry on oeis.org

4, 8, 9, 14, 15, 19, 22, 28, 29, 39, 42, 43, 44, 49, 53, 59, 60, 64, 65, 67, 74, 75, 78, 80, 82, 84, 85, 93, 94, 98, 99, 108, 109, 113, 114, 117, 120, 124, 127, 129, 133, 140, 144, 148, 152, 155, 157, 158, 159, 162, 163, 164, 169, 183, 184, 185, 194, 197, 198, 199

Offset: 1

Lekraj Beedassy, Aug 20 2006

nonn

Entries of A024898 which are not in A002822 or equivalently, entries of A046954 which are not in A060461.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A121762, A121765.

Filtered([1..250], k-> IsPrime(6*k-1) and not IsPrime(6*k+1)); # G. C. Greubel, Feb 20 2019

[n: n in [1..250] | IsPrime(6*n-1) and not IsPrime(6*n+1)]; // G. C. Greubel, Feb 20 2019

Select[Range[200], PrimeQ[6# -1] && !PrimeQ[6# +1] &] (* Ray Chandler, Aug 22 2006 *)

for(n=1, 250, if(isprime(6*n-1) && !isprime(6*n+1), print1(n", "))) \\ G. C. Greubel, Feb 20 2019

[n for n in (1..250) if is_prime(6*n-1) and not is_prime(6*n+1)] # G. C. Greubel, Feb 20 2019

Extended by Ray Chandler, Aug 22 2006

cp's OEIS Frontend

A218829 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(m)) + 2 are both prime.

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A124518 Numbers k such that 10k-1 and 10k+1 are twin primes.

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A236531 a(n) = |{0 < k < n: {6*k -1 , 6*k + 1} and {prime(n-k), prime(n-k) + 2} are both twin prime pairs}|.

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A060213 Lesser of twin primes whose average is 6 times a prime.

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A124065 Numbers k such that 8*k - 1 and 8*k + 1 are twin primes.

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A121765 Numbers n such that 6*n-1 is composite while 6*n+1 is prime.

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A182481 a(n) is the least k such that 6*k*n-1 and 6*k*n+1 are twin primes, and a(n)=0, if such k does not exist.

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A236531 a(n) = |{0 < k < n: {6k -1 , 6k + 1} and {prime(n-k), prime(n-k) + 2} are both twin prime pairs}|.

A124065 Numbers k such that 8k - 1 and 8k + 1 are twin primes.

A121765 Numbers n such that 6n-1 is composite while 6n+1 is prime.

A182481 a(n) is the least k such that 6kn-1 and 6kn+1 are twin primes, and a(n)=0, if such k does not exist.

A121763 Numbers n such that 6n-1 is prime while 6n+1 is composite.