A236508 -id:A236508 - cp's OEIS frontend

A236509 Primes p with p + 2, p + 6 and prime(p) + 6 all prime.

5, 11, 107, 227, 311, 347, 821, 857, 1091, 1607, 1997, 2657, 3527, 4931, 5231, 8087, 8231, 9431, 10331, 11171, 12917, 13691, 13877, 21377, 22271, 24917, 27737, 29567, 32057, 33347, 35591, 36467, 37307, 39227, 42017

Offset: 1

Zhi-Wei Sun, Jan 27 2014

nonn

According to the conjecture in A236508, this sequence should have infinitely many terms.

			a(1) = 5 since 5, 5 + 2 = 7, 5 + 6 = 11 and prime(5) + 6 = 17 are all prime, but 2 + 2 = 4 and 3 + 6 = 9 are both composite.

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

Cf. A000040, A022004, A023201, A046117, A236464, A236508.

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

A236511 a(n) = |{0 < k < n: p = 3*phi(k) + phi(n-k) - 1, p + 2, p + 6 and p + 8 are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 27 2014

nonn

Conjecture: a(n) > 0 for all n > 1075.
We have verified this for n up to 50000.
The above conjecture implies the well-known conjecture that there are infinitely many prime quadruplets (p, p + 2, p + 6, p + 8).

			a(10) = 1 since 3*phi(3) + phi(7) - 1 = 6 + 6 - 1 = 11, 11 + 2 = 13, 11 + 6 = 17 and 11 + 8 = 19 are all prime.
a(57) = 1 since 3*phi(31) + phi(26) - 1 = 90 + 12 - 1 = 101, 101 + 2 = 103, 101 + 6 = 107 and 101 + 8 = 109 are all prime.

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

Cf. A000010, A000040, A007530, A236508.

p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[n+6]&&PrimeQ[n+8]
f[n_,k_]:=3*EulerPhi[k]+EulerPhi[n-k]-1
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A236509 Primes p with p + 2, p + 6 and prime(p) + 6 all prime.

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