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

1,14

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.

Links

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A076473 A163160 A306696 * A029240 A302642 A025803

Adjacent sequences: A237181 A237182 A237183 * A237185 A237186 A237187

Keyword

nonn

Author

Zhi-Wei Sun, Feb 04 2014

Status

approved