A236462 -id:A236462 - cp's OEIS frontend

A236464 Primes p with prime(p) + 2 and prime(p) + 6 both prime.

3, 5, 7, 13, 43, 89, 313, 613, 643, 743, 1171, 1279, 1627, 1823, 1867, 1999, 2311, 2393, 2683, 2753, 2789, 3571, 4441, 4561, 5039, 5231, 5647, 5953, 6067, 6317, 6899, 8039, 8087, 8753, 8923, 9337, 9787, 9931, 10259, 10667

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the conjecture in A236472, this sequence contains infinitely many terms, i.e., there are infinitely many prime triples of the form {prime(p), prime(p) + 2, prime(p) + 6} with p prime.

See A236462 for a similar sequence.

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

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

Cf. A000040, A022004, A236457, A236458, A236462, A236472.

p[n_]:=p[n]=PrimeQ[Prime[n]+2]&&PrimeQ[Prime[n]+6]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]

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

Original entry on oeis.org

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 *)

A236460 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 - 1, prime(p) + 4 and prime(p) + 6 are all prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 1, 2, 2, 2, 1, 1, 2, 0, 4, 1, 2, 1, 5, 2, 1, 3, 1, 1, 3, 2, 6, 3, 0, 2, 5, 5, 6, 3, 4, 5, 3, 4, 4, 4, 6, 3, 2, 6, 2, 3, 2, 10, 2, 3, 1, 6, 1, 4, 0, 2, 3, 4, 2, 4, 0, 4, 0, 3, 2, 3, 0, 4, 0, 1, 1, 4

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

Conjecture: a(n) > 0 for all n > 211.

This implies that there are infinitely many primes p with {prime(p), prime(p) + 4, prime(p) + 6} a prime triple. See A236462 for such primes p.

			a(30) = 1 since 30 = 13 + 17 with phi(13) + phi(17)/2 - 1 = 19, prime(19) + 4 = 67 + 4 = 71 and prime(19) + 6 = 73 all prime.
a(831) = 1 since 831 = 66 + 765 with phi(66) + phi(765)/2 - 1 = 20 + 192 - 1 = 211, prime(211) + 4 = 1297 + 4 = 1301 and prime(211) + 6 = 1303 all prime.

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

Cf. A000010, A000040, A001359, A006512, A022005, A236456, A236457, A236458, A236462, A236464.

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+4]&&PrimeQ[Prime[n]+6]
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}]

A236552 a(n) = |{0 < k < n: 6k - 1, 6k + 1, 6*k + 5 and prime(n-k) + 6 are all prime}|.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 4, 5, 4, 4, 3, 4, 3, 5, 5, 5, 2, 2, 4, 3, 4, 3, 3, 4, 5, 3, 5, 4, 6, 6, 6, 4, 4, 1, 3, 6, 5, 2, 4, 4, 5, 6, 8, 6, 5, 4, 8, 4, 4, 5, 7, 6, 4, 4, 4, 6, 6, 5, 5, 6, 6, 7, 6, 4, 3, 5, 7, 3, 4, 4, 3, 6, 5, 5, 6, 6, 7, 5, 4, 3, 5, 5, 3, 3, 2, 2

Offset: 1

Zhi-Wei Sun, Jan 28 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 3.
(ii) For any integer n > 4, there is a positive integer k < n such that 6*k - 5, 6*k - 1, 6*k + 1 and prime(n-k) + 6 are all prime.
(iii) Any integer n > 7 can be written as p + q with q > 0 such that p, p + 6 and prime(q) + 6 are all prime.
(iv) Each integer n > 4 can be written as k*(k+1) + m with k > 0 and m > 0 such that prime(m) + 6 is prime.

			a(4) = 1 since 6*1 - 1, 6*1 + 1, 6*1 + 5 and prime(4-1) + 6 = 11 are all prime.
a(48) = 1 since 6*32 - 1 = 191, 6*32 + 1 = 193, 6*32 + 5 = 197 and prime(48-32) + 6 = 53 + 6 = 59 are all prime.

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

Cf. A000040, A022004, A022005, A023201, A046117, A236462, A236464.

p[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]&&PrimeQ[6n+5]
q[n_]:=PrimeQ[Prime[n]+6]
a[n_]:=Sum[If[p[k]&&q[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

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A236464 Primes p with prime(p) + 2 and prime(p) + 6 both prime.

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

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A236460 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 - 1, prime(p) + 4 and prime(p) + 6 are all prime, where phi(.) is Euler's totient function.

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A236552 a(n) = |{0 < k < n: 6*k - 1, 6*k + 1, 6*k + 5 and prime(n-k) + 6 are all prime}|.

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A236552 a(n) = |{0 < k < n: 6k - 1, 6k + 1, 6*k + 5 and prime(n-k) + 6 are all prime}|.