A236460 -id:A236460 - cp's OEIS frontend

A236462 Primes p with prime(p) + 4 and prime(p) + 6 both prime.

19, 59, 151, 181, 211, 229, 389, 571, 877, 983, 1039, 1259, 1549, 3023, 3121, 3191, 3259, 3517, 3719, 4099, 4261, 4463, 5237, 6947, 7529, 7591, 7927, 7933, 8317, 8389, 8971, 9403, 9619, 10163, 10939, 11131, 11717, 11743, 11839, 12301

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the conjecture in A236460, this sequence should have infinitely many terms.

See A236464 for a similar sequence.

			a(1) = 19 with 19, prime(19) + 4 = 71 and prime(19) + 6 = 73 all prime.

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

Cf. A000040, A022005, A236456, A236457, A236458, A236460, A236464.

p[n_]:=p[n]=PrimeQ[Prime[n]+4]&&PrimeQ[Prime[n]+6]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]
Select[Prime[Range[1500]],AllTrue[Prime[#]+{4,6},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 21 2018 *)

s=[]; forprime(p=2, 12500, if(isprime(prime(p)+4) && isprime(prime(p)+6), 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}]

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}]

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A236462 Primes p with prime(p) + 4 and prime(p) + 6 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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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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