A235343 a(n) = |{0 < k < n: f(n,k) - 1, f(n,k) + 1 and q(f(n,k)) + 1 are all prime with f(n,k) = phi(k) + phi(n-k)/4}|, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).

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

Offset: 1

Author

Zhi-Wei Sun, Jan 06 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n >= 60.

(ii) For any integer n > 1234, there is a positive integer k < n such that g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime, where g(n,k) = phi(k) + phi(n-k)/8.

Clearly, part (i) implies that there are infinitely many primes of the form q(m) + 1 with m - 1 and m + 1 also prime, and part (ii) implies that there are infinitely many primes of the form q(m) - 1 with m - 1 and m + 1 also prime. As log q(m) is asymptotically equivalent to pi*sqrt(m/3), the conjecture is much stronger than the twin prime conjecture.

We have verified parts (i) and (ii) for n up to 100000 and 60000 respectively.

Examples

			a(50) = 1 since phi(34) + phi(16)/4 = 18 with 18 - 1, 18 + 1 and q(18) + 1 = 47 all prime.
a(215) = 1 since phi(87) + phi(128)/4 = 72 with 72 - 1, 72 + 1 and q(72) + 1 = 36353 all prime.
a(645) = 1 since phi(365) + phi(280)/4 = 312 with 312 - 1, 312 + 1 and q(312) + 1 = 207839472391 all prime.
		

Links

• 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

Crossrefs

Cf. A000009, A000010, A000040, A001359, A006512, A014574, A234514, A234567, A234615, A235344, A235346, A235356, A235357, A235358.

Programs

Mathematica

f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/4
p[n_,k_]:=PrimeQ[f[n,k]-1]&&PrimeQ[f[n,k]+1]&&PrimeQ[PartitionsQ[f[n,k]]+1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]