A236074 -id:A236074 - cp's OEIS frontend

A236097 a(n) = |{0 < k < n-2: p = phi(k) + phi(n-k)/2 + 1, prime(p) - p - 1 and prime(p) - p + 1 are all prime}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 3, 1, 1, 2, 2, 3, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 0, 5, 5, 2, 4, 1, 5, 3, 3, 2, 4, 4, 9, 5, 9, 4, 10, 3, 6, 6, 8, 5, 10, 4, 4, 7, 8, 10, 5, 8, 9, 9, 4, 11, 3, 5, 5, 9, 5, 4, 4, 5, 6, 8, 7, 6, 3, 11, 4, 8, 10, 9, 8, 7, 6, 11, 7, 9, 4, 6, 5, 6, 2, 9, 4, 7, 6, 7, 10, 9

Offset: 1

Zhi-Wei Sun, Jan 19 2014

nonn

Conjecture: a(n) > 0 for all n > 31.

This implies that there are infinitely many primes p with {prime(p) - p - 1, prime(p) - p + 1} a twin prime pair.

			a(20) = 1 since phi(2) + phi(18)/2 + 1 = 5, prime(5) - 5 - 1 = 5 and prime(5) - 5 + 1 = 7 are all prime.
a(36) = 1 since phi(21) + phi(15)/2 + 1 = 17, prime(17) - 17 - 1 = 41 and prime(17) - 17 + 1 = 43 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000010, A000040, A001359, A006512, A014574, A234694, A234695, A235924, A236074, A236119.

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n-1]&&PrimeQ[Prime[n]-n+1]
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}]

A236075 Odd primes p with prime(2p) - 2prime(p) and prime(p) - 2*prime((p-1)/2) both prime.

Original entry on oeis.org

5, 29, 79, 101, 103, 109, 353, 487, 821, 1367, 1811, 2111, 2593, 2617, 2939, 2969, 3001, 3659, 3727, 3877, 3911, 5347, 5779, 6481, 6959, 7121, 9059, 9649, 10007, 10099, 11299, 11311, 11827, 12343, 12511, 12539, 12589, 12689, 12923, 13781, 13967, 14249, 15859, 15923, 16363, 16889, 17321, 17683, 17881, 18181

Offset: 1

Zhi-Wei Sun, Jan 19 2014

nonn

By the conjecture in A236074, this sequence should have infinitely many terms.

			a(1) = 5 since neither prime(2*2) - 2*prime(2) = 1 nor  prime(3) - 2*prime((3-1)/2) = 1 is prime, but prime(2*5) - 2*prime(5) = 29 - 2*11 = 7 and prime(5) - 2*prime((5-1)/2) = 11 - 2*3 = 5 are both prime.

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

Cf. A000040, A236074.

PQ[n_]:=n>0&&PrimeQ[n]
p[n_]:=PQ[Prime[2n]-2Prime[n]]&&PQ[Prime[n]-2*Prime[(n-1)/2]]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,2,10^6}]

s=[]; forprime(p=3, 20000, if(isprime(prime(2*p)-2*prime(p)) && isprime(prime(p)-2*prime((p-1)/2)), s=concat(s, p))); s \\ Colin Barker, Jan 19 2014

A236138 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/3 - 1, prime(p-1) - (p-1) and prime(p-1) - 2*prime((p-1)/2) 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, 1, 0, 2, 0, 3, 0, 1, 3, 1, 4, 0, 5, 2, 2, 3, 4, 1, 4, 2, 5, 3, 1, 6, 4, 3, 0, 4, 5, 3, 3, 4, 5, 2, 4, 2, 2, 4, 3, 4, 1, 2, 2, 3, 5, 3, 0, 3, 2, 4, 1, 2, 2, 4, 0, 4, 1, 3, 3, 2, 0, 4, 1, 3, 2, 3, 1, 5, 3, 5, 1, 4, 2, 3, 5, 4, 4, 5, 4, 1, 2, 2, 3, 3, 7, 3, 2, 3

Offset: 1

Zhi-Wei Sun, Jan 19 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 73.
(ii) For any integer n > 472, there is a positive integer k < n such that p = phi(k) + phi(n-k)/4 - 1, prime(p) - 2*prime((p-1)/2) and prime(p) - 2*prime((p+1)/2) are all prime.
Clearly, part (i) of the conjecture implies that there are infinitely many odd primes p with prime(p-1) - (p-1) and prime(p-1) - 2*prime((p-1)/2) both prime, and part (ii) implies that there are infinitely many odd primes p with prime(p) - 2*prime((p-1)/2) and prime(p) - 2*prime((p+1)/2) both prime.

			a(20) = 1 since phi(11) + phi(9)/3 - 1 = 11, prime(10) - 10 = 29 - 10 = 19 and prime(10) - 2*prime(5) = 29 - 2*11 = 7 are all prime.
a(293) = 1 since phi(267) + phi(26)/3 - 1 = 176 + 12/3 - 1 = 179, prime(178) - 178 = 1061 - 178 = 883 and prime(178) - 2*prime(89) = 1061 - 2*461 = 139 are all prime.

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

Cf. A000010, A000040, A234694, A235924, A236074, A236097.

PQ[n_]:=n>0&&PrimeQ[n]
p[n_]:=n>2&&PrimeQ[n]&&PrimeQ[Prime[n-1]-(n-1)]&&PQ[Prime[n-1]-2*Prime[(n-1)/2]]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/3-1
  a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A236097 a(n) = |{0 < k < n-2: p = phi(k) + phi(n-k)/2 + 1, prime(p) - p - 1 and prime(p) - p + 1 are all prime}|, where phi(.) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A236075 Odd primes p with prime(2p) - 2prime(p) and prime(p) - 2*prime((p-1)/2) both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A236138 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/3 - 1, prime(p-1) - (p-1) and prime(p-1) - 2*prime((p-1)/2) are all prime}|, where phi(.) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

cp's OEIS Frontend

A236097 a(n) = |{0 < k < n-2: p = phi(k) + phi(n-k)/2 + 1, prime(p) - p - 1 and prime(p) - p + 1 are all prime}|, where phi(.) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A236075 Odd primes p with prime(2*p) - 2*prime(p) and prime(p) - 2*prime((p-1)/2) both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A236138 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/3 - 1, prime(p-1) - (p-1) and prime(p-1) - 2*prime((p-1)/2) are all prime}|, where phi(.) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A236075 Odd primes p with prime(2p) - 2prime(p) and prime(p) - 2*prime((p-1)/2) both prime.