A237184 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.

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, 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, 4, 5, 2, 7, 2, 5, 4, 4, 3, 6, 2, 5

Offset: 1

Author

Zhi-Wei Sun, Feb 04 2014

Keywords

• nonn

Comments

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

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

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

Examples

			a(10) = 1 since 10 = 7 + 3 with 7, 3, phi(7+1) - 1 = 3 and phi(3-1) + 1 = 2 all prime.
a(499) = 1 since 499 = 2*199 + 101 with 199, 101, phi(199+1) - 1 = 79 and phi(101-1) + 1 = 41 all prime.
a(869) = 1 since 869 = 2*433 + 3 with 433, 3, phi(433+1) - 1 = 179 and phi(3-1) + 1 = 2 all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

pq[n_]:=PrimeQ[n]&&PrimeQ[EulerPhi[n+1]-1]
PQ[n_]:=PrimeQ[n]&&PrimeQ[EulerPhi[n-1]+1]
a[n_]:=Sum[If[pq[k]&&PQ[n-(1+Mod[n,2])k],1,0],{k,1,(n-1)/(1+Mod[n,2])}]
Table[a[n],{n,1,80}]