A236467 -id:A236467 - cp's OEIS frontend

A236458 Primes p with p + 2 and prime(p) + 2 both prime.

3, 5, 17, 41, 1949, 2309, 2711, 2789, 2801, 3299, 3329, 3359, 3917, 4157, 4217, 4259, 4637, 5009, 5021, 5231, 6449, 7757, 8087, 8219, 8627, 9419, 9929, 10007, 10937, 11777, 12071, 14321, 15647, 15971, 16061, 16901, 18131, 18251, 18287, 18539

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the conjecture in A236470, the sequence should have infinitely many terms. This is stronger than the twin prime conjecture.

See A236457 and A236467 for similar sequences.

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

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

Cf. A000040, A001359, A006512, A236119, A236456, A236457, A236467, A236470.

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}]

s=[]; forprime(p=2, 20000, if(isprime(p+2) && isprime(prime(p)+2), s=concat(s, p))); s \\ Colin Barker, Jan 26 2014

A236468 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 - 1, p + 2 and 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, 2, 2, 1, 3, 1, 1, 2, 2, 4, 0, 1, 2, 2, 1, 2, 1, 1, 2, 0, 3, 2, 2, 3, 4, 2, 1, 2, 5, 3, 4, 0, 6, 6, 1, 3, 1, 5, 4, 5, 2, 5, 1, 7, 1, 3, 2, 5, 1, 4, 1, 7, 0, 5, 4, 1, 8, 1, 5, 5, 1, 2, 5, 4, 4, 4, 4, 1, 5, 1, 7, 3, 3, 2, 2, 1, 8, 3, 3, 2, 2, 2, 6, 3, 7, 2, 6, 5, 1, 1, 5, 4, 9, 3

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

Conjecture: a(n) > 0 for every n = 250, 251, ....

This implies that there are infinitely many twin prime pairs {p, p + 2} with {prime(p) - 2, prime(p)} also a twin prime pair. It is stronger than the twin prime conjecture.

			 a(33) = 1 since 33 = 7 + 26 with phi(7) + phi(26)/2 - 1 = 11, 11 + 2 = 13 and prime(11) - 2 = 31 - 2 = 29 all prime.
a(278) = 1 since 278 = 61 + 217 with phi(61) + phi(217)/2 - 1 = 60 + 90 - 1 = 149, 149 + 2 = 151 and prime(149) - 2 = 859 - 2 = 857 all prime.

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

Cf. A000010, A000040, A001359, A006512, A236097, A236456, A236460, A236467.

p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n]-2]
f[n_,k_]:=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}]

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

Original entry on oeis.org

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}]

A236472 a(n) = |{0 < k < n: p = prime(k) + phi(n-k), prime(p) + 2 and prime(p) + 6 are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 0, 1, 1, 2, 3, 0, 1, 1, 1, 2, 0, 1, 1, 0, 0, 2, 0, 2, 2, 1, 0, 0, 3, 1, 2, 0, 2, 2, 2, 1, 0, 0, 4, 1, 0, 0, 0, 0, 5, 0, 1, 1, 1, 2, 1, 1, 3, 0, 0, 2, 2, 0, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 3, 2, 0, 0, 2, 1, 1, 3, 0, 0, 2, 0, 3, 0, 0, 1, 1, 0, 2, 0, 0

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

Conjecture: a(n) > 0 for every n = 330, 331, ....
We have verified this for n up to 80000.
The conjecture implies that there are infinitely many prime triples of the form {prime(p), prime(p) + 2, prime(p) + 6} with p prime. See A236464 for such primes p.

			a(10) = 1 since prime(2) + phi(8) = 3 + 4 = 7, prime(7) + 2 = 17 + 2 = 19 and prime(7) + 6 = 23 are all prime.
a(877) = 1 since prime(784) + phi(877-784) = 6007 + 60 = 6067, prime(6067) + 2 = 60101 + 2 = 60103 and prime(6067) + 6 = 60107 are all prime.

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

Cf. A000010, A000040, A022004, A236456, A236460, A236464, A236467, A236470.

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[n]+6]
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}]

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 *)

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A236458 Primes p with p + 2 and prime(p) + 2 both prime.

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A236468 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 - 1, p + 2 and prime(p) - 2 are all prime, where phi(.) is Euler's totient function.

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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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A236472 a(n) = |{0 < k < n: p = prime(k) + phi(n-k), prime(p) + 2 and prime(p) + 6 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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