A237184 - OEIS

%I #14 Feb 04 2014 09:03:09

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

%T 1,5,3,1,2,4,3,5,2,3,4,4,1,7,3,4,4,4,2,6,2,5,4,4,2,7,3,2,4,5,3,8,2,2,

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

%N Number of ordered ways to write n = (1+(n mod 2))*p + q with p, q, phi(p+1) - 1 and phi(q-1) + 1 all prime.

%C Conjecture: a(n) > 0 for all n > 23.

%C This is stronger than Goldbach's conjecture and Lemoine's conjecture (cf. A046927).

%C We have verified the conjecture for n up to 3*10^6.

%H Zhi-Wei Sun, <a href="/A237184/b237184.txt">Table of n, a(n) for n = 1..10000</a>

%e a(10) = 1 since 10 = 7 + 3 with 7, 3, phi(7+1) - 1 = 3 and phi(3-1) + 1 = 2 all prime.

%e a(499) = 1 since 499 = 2*199 + 101 with 199, 101, phi(199+1) - 1 = 79 and phi(101-1) + 1 = 41 all prime.

%e a(869) = 1 since 869 = 2*433 + 3 with 433, 3, phi(433+1) - 1 = 179 and phi(3-1) + 1 = 2 all prime.

%t pq[n_]:=PrimeQ[n]&&PrimeQ[EulerPhi[n+1]-1]

%t PQ[n_]:=PrimeQ[n]&&PrimeQ[EulerPhi[n-1]+1]

%t a[n_]:=Sum[If[pq[k]&&PQ[n-(1+Mod[n,2])k],1,0],{k,1,(n-1)/(1+Mod[n,2])}]

%t Table[a[n],{n,1,80}]

%Y Cf. A000040, A002372, A002375, A039698, A046927, A078892, A237127, A237130, A237168, A237183.

%K nonn

%O 1,14

%A _Zhi-Wei Sun_, Feb 04 2014