A236468 -id:A236468 - cp's OEIS frontend

A236467 Primes p with p + 2 and prime(p) - 2 both prime.

3, 11, 29, 149, 179, 191, 269, 347, 431, 461, 617, 659, 1031, 1619, 1931, 3467, 3527, 4799, 6569, 6689, 7349, 7877, 9011, 9767, 11117, 12611, 13691, 13901, 14549, 16067, 16139, 16451, 16631, 17489, 17681, 18911, 20981, 22367, 23909, 24179

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the conjecture in A236468, this sequence should have infinitely many terms.

See A236457 and A236458 for similar sequences.

			a(1) = 3 since 3, 3 + 2 = 5 and prime(3) - 2 = 3 are all prime, but 2 + 2 = 4 is composite.

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

Cf. A000040, A001359, A006512, A236119, A236457, A236458, A236462, A236464, A236468.

p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[Prime[n]-2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]
Select[Prime[Range[3000]],AllTrue[{#+2,Prime[#]-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 11 2020 *)

A236470 a(n) = |{0 < k < n: p = prime(k) + phi(n-k), p + 2 and prime(p) + 2 are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

Conjecture: a(n) > 0 for all n > 948.
We have verified this for n up to 50000.
The conjecture implies that there are infinitely many primes p with p + 2 and prime(p) + 2 both prime. See A236458 for such primes p.

			 a(12) = 1 since prime(5) + phi(7) = 11 + 6 = 17, 17 + 2 = 19 and prime(17) + 2 = 59 + 2 = 61 are all prime.
a(97) = 1 since prime(7) + phi(90) = 17 + 24 = 41, 41 + 2 = 43 and prime(41) + 2 = 179 + 2 = 181 are all prime.

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

Cf. A000010, A000040, A001359, A006512, A236097, A236456, A236458, A236468.

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

A236481 Primes p with p + 2, prime(p) + 2 and prime(prime(p)) + 2 all prime.

Original entry on oeis.org

3, 1949, 4217, 8219, 9929, 22091, 23537, 28097, 38711, 41609, 50051, 60899, 68111, 72227, 74159, 79631, 115151, 122399, 127679, 150959, 155537, 266687, 267611, 270551, 271499, 284741, 306347, 428297, 433661, 444287

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

Conjecture: For any positive integer m, there are infinitely many chains p(1) < p(2) < ... < p(m) of m primes with p(k) + 2 prime for all k = 1,...,m such that p(k + 1) = prime(p(k)) for every 0 < k < m.

			a(1) = 3 since 3, 3 + 2 = 5, prime(3) + 2 = 7 and prime(prime(3)) + 2 = prime(5) + 2 = 13 are all prime, but 2 + 2 = 4 is composite.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, A new kind of prime chains, a message to Number Theory List, Jan. 20, 2014.

Cf. A000010, A000040, A001359, A006512, A235925, A236066, A236457, A236458, A236468, A236470, A236480, A236482, A236484.

p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[Prime[n]+2]&&PrimeQ[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}]

cp's OEIS Frontend

A236467 Primes p with p + 2 and prime(p) - 2 both prime.

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A236470 a(n) = |{0 < k < n: p = prime(k) + phi(n-k), p + 2 and prime(p) + 2 are all prime}|, where phi(.) is Euler's totient function.

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A236481 Primes p with p + 2, prime(p) + 2 and 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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