|
%I #6 Feb 05 2014 08:49:45
%S 0,0,0,0,0,1,0,2,1,2,2,1,2,1,2,1,1,1,2,2,3,4,2,2,1,2,3,3,3,2,4,5,4,3,
%T 4,3,5,4,4,6,6,7,5,5,6,3,4,3,6,5,6,5,3,6,5,6,3,3,5,3,5,4,3,4,3,6,4,3,
%U 1,1,4,3,4,4,4,5,6,7,3,3
%N Number of ordered ways to write n = k + m with k > 0 and m > 0 such that phi(k) - 1, phi(k) + 1 and prime(prime(prime(m))) - 2 are all prime, where phi(.) is Euler's totient function.
%C Conjecture: (i) a(n) > 0 for all n > 7.
%C (ii) Any integer n > 22 can be written as k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(prime(m))) - 2 are both prime.
%C Note that either part of the conjecture implies the twin prime conjecture.
%H Zhi-Wei Sun, <a href="/A237253/b237253.txt">Table of n, a(n) for n = 1..10000</a>
%e a(12) = 1 since 12 = 9 + 3 with phi(9) - 1 = 5, phi(9) + 1 = 7 and prime(prime(prime(3))) - 2 = prime(prime(5)) - 2 = prime(11) - 2 = 29 all prime.
%e a(103) = 1 since 103 = 73 + 30 with phi(73) - 1 = 71, phi(73) + 1 = 73 and prime(prime(prime(30))) - 2 = prime(prime(113)) - 2 = prime(617) - 2 = 4547 all prime.
%t pq[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
%t PQ[n_]:=PrimeQ[Prime[Prime[Prime[n]]]-2]
%t a[n_]:=Sum[If[pq[k]&&PQ[n-k],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000010, A000040, A001359, A006512, A072281, A218829, A236531, A236566, A237127, A237130, A237168.
%K nonn
%O 1,8
%A _Zhi-Wei Sun_, Feb 05 2014
|
