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Number of ways to write 2*n - 1 = 2*p + q with p, q, phi(p+1) - 1 and phi(p+1) + 1 all prime, where phi(.) is Euler's totient function.
6

%I #8 Feb 04 2014 08:58:52

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

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

%U 2,2

%N Number of ways to write 2*n - 1 = 2*p + q with p, q, phi(p+1) - 1 and phi(p+1) + 1 all prime, where phi(.) is Euler's totient function.

%C Conjecture: (i) a(n) > 0 for all n > 12.

%C (ii) Any even number greater than 4 can be written as p + q with p, q, phi(p+2) - 1 and phi(p+2) + 1 all prime.

%C Part (i) implies both Lemoine's conjecture (cf. A046927) and the twin prime conjecture, while part (ii) unifies Goldbach's conjecture and the twin prime conjecture.

%H Zhi-Wei Sun, <a href="/A237168/b237168.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;b634e7c8.1401">Unification of Goldbach's conjecture and the twin prime conjecture</a>, a message to Number Theory List, Jan. 29, 2014.

%e a(9) = 1 since 2*9 - 1 = 2*7 + 3 with 7, 3, phi(7+1) - 1 = 3 and phi(7+1) + 1 = 5 all prime.

%e a(934) = 1 since 2*934 - 1 = 2*457 + 953 with 457, 953, phi(457+1) - 1 = 227 and phi(457+1) + 1 = 229 all prime.

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

%t a[n_]:=Sum[If[PQ[Prime[k]+1]&&PrimeQ[2n-1-2*Prime[k]],1,0],{k,1,PrimePi[n-1]}]

%t Table[a[n],{n,1,70}]

%Y Cf. A000010, A000040, A001359, A002375, A002372, A006512, A046927, A072281, A236566, A237127, A237130.

%K nonn

%O 1,13

%A _Zhi-Wei Sun_, Feb 04 2014