A235343 - OEIS

%I #19 Apr 06 2014 04:08:02

%S 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,

%T 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,

%U 1,3,6,3,3,1,5,2,2,2,4,2,2,2,2,2,3,2,2

%N 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).

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

%C (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.

%C 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.

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

%H Zhi-Wei Sun, <a href="/A235343/b235343.txt">Table of n, a(n) for n = 1..10000</a>

%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014

%e a(50) = 1 since phi(34) + phi(16)/4 = 18 with 18 - 1, 18 + 1 and q(18) + 1 = 47 all prime.

%e a(215) = 1 since phi(87) + phi(128)/4 = 72 with 72 - 1, 72 + 1 and q(72) + 1 = 36353 all prime.

%e a(645) = 1 since phi(365) + phi(280)/4 = 312 with 312 - 1, 312 + 1 and q(312) + 1 = 207839472391 all prime.

%t f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/4

%t p[n_,k_]:=PrimeQ[f[n,k]-1]&&PrimeQ[f[n,k]+1]&&PrimeQ[PartitionsQ[f[n,k]]+1]

%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]

%t Table[a[n],{n,1,100}]

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

%K nonn

%O 1,19

%A _Zhi-Wei Sun_, Jan 06 2014