A236458 -id:A236458 - cp's OEIS frontend

A236482 Primes p with p + 2, prime(p) + 2, prime(prime(p)) + 2 and prime(prime(prime(p))) + 2 all prime.

41609, 1119047, 1928621, 2348579, 2371709, 3406727, 4098569, 4204817, 4438997, 5561819, 6161159, 6293297, 8236439, 8736701, 8890667, 8951387, 9231329, 9390077, 10492457, 10619897, 11255729, 11514719, 11769479, 11920661, 12316697

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the general conjecture in A236481, this sequence should have infinitely many terms.

			a(1) = 41609 with 41609, 41609 + 2 = 41611, prime(41609) + 2 = 500909 + 2 = 500911, prime(500909) + 2 = 7382957 + 2 = 7382959 and prime(7382957) + 2 = 130090109 + 2 = 130090111 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100

Cf. A000040, A001359, A006512, A236457, A236458, A236467, A236480, A236481, A236484.

p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[Prime[n]]+2]&&PrimeQ[Prime[Prime[Prime[n]]]+2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10^6}]

A236480 a(n) = |{0 < k < n-2: p = 2*phi(k) + phi(n-k)/2 + 1, prime(p) + 2 and prime(prime(p)) + 2 are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 2, 1, 3, 2, 2, 0, 2, 3, 1, 2, 1, 3, 3, 2, 2, 1, 1, 1, 3, 0, 2, 3, 2, 1, 3, 0, 2, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

Conjecture: a(n) > 0 for every n = 640, 641, ....
We have verified this for n up to 75000.
The conjecture implies that there are infinitely many primes p with prime(p) + 2 and prime(prime(p)) + 2 both prime.

			a(8) = 1 since 2*phi(3) + phi(5)/2 + 1 = 7, prime(7) + 2 = 17 + 2 = 19 and prime(prime(7)) + 2 = prime(17) + 2 = 61 are all prime.
a(667) = 1 since 2*phi(193) + phi(667-193)/2 + 1 = 384 + 78 + 1 = 463, prime(463) + 2 = 3299 + 2 = 3301 and prime(prime(463)) + 2 = prime(3299) + 2 = 30559 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A001359, A006512, A236456, A236457, A236458, A236468, A236470, A236481.

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[Prime[n]]+2]
f[n_,k_]:=2*EulerPhi[k]+EulerPhi[n-k]/2+1
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

A237283 Primes p with prime(prime(p)) + 2 also prime.

Original entry on oeis.org

2, 3, 7, 13, 23, 29, 59, 71, 103, 193, 257, 271, 281, 311, 317, 389, 433, 439, 463, 569, 577, 619, 673, 683, 691, 797, 811, 857, 859, 887, 1031, 1069, 1109, 1129, 1153, 1229, 1307, 1597, 1613, 1867, 1949, 1951, 2069, 2297, 2477, 2551, 2621, 2657, 2699, 2753

Offset: 1

Zhi-Wei Sun, Feb 05 2014

nonn

This sequence is interesting because of the conjecture in A237253.

A236481, A236482 and A236484 are subsequences of the sequence.

			a(1) = 2 since 2 and prime(prime(2)) + 2 = prime(3) + 2 = 7 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A001359, A006512, A096478, A218829, A236457, A236458, A236467, A236481, A236482, A236484, A236568, A237253.

n=0;Do[If[PrimeQ[Prime[Prime[Prime[k]]]+2],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]
Select[Prime[Range[500]],PrimeQ[Prime[Prime[#]]+2]&] (* Harvey P. Dale, May 30 2018 *)

A259488 Positive integers k with prime(k)+2 and prime(prime(k))+2 both prime.

Original entry on oeis.org

2, 3, 7, 13, 296, 343, 395, 405, 408, 463, 469, 473, 542, 572, 577, 584, 625, 671, 673, 695, 837, 984, 1016, 1030, 1074, 1165, 1224, 1230, 1328, 1410, 1445, 1679, 1825, 1860, 1867, 1949, 2078, 2091, 2095, 2123, 2167, 2476, 2478, 2616, 2753, 2764, 2956, 3011, 3065, 3416, 3621, 3646, 3712, 3720, 3758, 3872, 3926, 4063, 4071, 4079, 4133, 4217, 4312, 4351, 4524, 4745, 4855, 4865, 4882, 4922

Offset: 1

Zhi-Wei Sun, Jun 28 2015

nonn

The conjecture in A259487 essentially says that {a(m)/a(n): m,n = 1,2,3,...} coincides with the set of all positive rational numbers. This implies that the current sequence has infinitely many terms.

			a(1) = 2 since prime(2)+2 = 5 and prime(prime(2))+2 = prime(3)+2 = 7 are both prime, but prime(1)+2 = 4 is composite.
a(2) = 3 since prime(3)+2 = 7 and prime(prime(3))+2 = prime(7)+2 = 19 are both prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

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

Cf. A000040, A001359, A006512, A236458, A236470, A259487.

n=0;Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]]+2],n=n+1;Print[n," ",k]],{k,1,5000}]
Select[Range[5000],AllTrue[{Prime[#]+2,Prime[Prime[#]]+2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 18 2018 *)

k=pk=0; forprime(ppk=2,1e6, if(isprime(pk++),k++; if(isprime(pk+2) && isprime(ppk+2), print1(k", ")))) \\ Charles R Greathouse IV, Jun 29 2015

A261533 Primes p such that p+2 is prime with prime(p+2)-prime(p)=6.

Original entry on oeis.org

3, 5, 59, 2789, 5231, 6947, 8087, 11717, 15269, 16229, 17207, 17909, 18059, 18131, 24917, 28751, 35279, 37307, 39227, 39239, 41201, 43787, 45821, 47741, 51869, 53087, 53609, 58439, 64577, 69857, 70919, 75707, 79631, 84869, 92381, 93479, 96179, 102197, 102929, 106187

Offset: 1

Zhi-Wei Sun, Aug 23 2015

nonn

The conjecture in A261528 implies that the current sequence has infinitely many terms.

Note that for each n > 2 the difference prime(n+2)-prime(n) is at least 6.

			a(1) = 3 since 3 and 3+2 = 5 are twin prime, and prime(5)-prime(3) = 11-5 = 6.
a(2) = 5 since 5 and 5+2 = 7 are twin prime, and prime(7)-prime(5) = 17-11 = 6.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A001359, A006512, A023201, A046117, A236458, A259488, A259539, A261528.

f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6
n=0;Do[If[PQ[k],n=n+1;Print[n," ",f[k]]],{k,1,10119}]
Select[Partition[Prime[Range[11000]],2,1],#[[2]]-#[[1]]==2&&Prime[#[[1]]+ 2]- Prime[#[[1]]]==6&][[All,1]] (* Harvey P. Dale, Apr 26 2020 *)

isok(i)=p=prime(i);isprime(p+2)&&prime(p+2)-prime(p)==6;
first(m)=my(v=vector(m));i=1;for(j=1,m,while(!isok(i),i++);v[j]=prime(i);i++);v; \\ Anders Hellström, Aug 23 2015

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A236482 Primes p with p + 2, prime(p) + 2, prime(prime(p)) + 2 and prime(prime(prime(p))) + 2 all prime.

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A236480 a(n) = |{0 < k < n-2: p = 2*phi(k) + phi(n-k)/2 + 1, prime(p) + 2 and prime(prime(p)) + 2 are all prime}|, where phi(.) is Euler's totient function.

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A237283 Primes p with prime(prime(p)) + 2 also prime.

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A259488 Positive integers k with prime(k)+2 and prime(prime(k))+2 both prime.

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A261533 Primes p such that p+2 is prime with prime(p+2)-prime(p)=6.

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