A234615 - OEIS

%I #16 Apr 05 2014 22:30:48

%S 0,0,0,0,1,1,0,2,2,2,4,3,5,4,2,6,6,6,5,4,5,6,4,6,5,5,2,4,5,6,5,7,4,6,

%T 6,8,3,3,6,7,7,4,4,4,4,7,7,3,3,4,4,6,5,4,5,5,7,1,3,4,7,5,5,6,3,7,11,5,

%U 4,5,4,7,6,4,2,7,9,7,5,5,6,5,10,7,4,3,4,6,3,4,9,5,3,5,6,5,3,6,2,7

%N Number of ways to write n = k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and q(p) - 1 are both prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

%C Conjecture: (i) a(n) > 0 for all n > 7.

%C (ii) Any integer n > 7 not equal to 15 can be written as k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and q(p) + 1 are both prime.

%C (iii) Any integer n > 83 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a square. Also, each integer n > 45 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a triangular number.

%C Clearly, part (i) of this conjecture implies that there are infinitely many primes p with q(p) - 1 also prime (cf. A234644).

%H Zhi-Wei Sun, <a href="/A234615/b234615.txt">Table of n, a(n) for n = 1..8000</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(6) = 1 since 6 = 2 + 4 with prime(2) + phi(4) = 5 and q(5) - 1 = 2 both prime.

%e a(58) = 1 since 58 = 12 + 46 with prime(12) + phi(46) = 59 and q(59) - 1 = 9791 both prime.

%e a(526) = 1 since 526 = 389 + 137 with prime(389) + phi(137) = 2819 and q(2819) - 1 = 326033386646595458662191828888146112979 both prime.

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

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

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

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

%Y Cf. A000009, A000010, A000040, A234475, A234514, A234530, A234567, A234569, A234572, A234644

%K nonn

%O 1,8

%A _Zhi-Wei Sun_, Dec 28 2013