A235987 a(n) = |{0 < k < n: p = prime(k) + phi(n-k) is a Sophie prime having 2 as a primitive root}|.

0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 1, 0, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 3, 1, 2, 2, 0, 4, 3, 3, 1, 3, 0, 2, 2, 2, 4, 1, 4, 2, 1, 4, 3, 4, 6, 1, 2, 5, 2, 4, 5, 4, 4, 1, 5, 4, 1, 4, 1, 3, 2, 3, 4, 2, 3, 5, 2, 4, 5, 2, 6, 2, 2, 5, 1, 4, 8, 4, 3, 3, 2, 3, 4, 5, 5, 3, 7, 2, 4, 3, 5, 6, 7, 6, 3, 4

Offset

1,12

Comments

Conjecture: (i) a(n) > 0 for all n > 37.

(ii) For any integer n > 7, there is a positive integer k < n such that p = phi(k) + phi(n-k)/2 - 1 is a prime having 2 as a primitive root.

Part (i) of the conjecture implies that there are infinitely many Sophie Germain primes p having 2 as a primitive root modulo p.

We also have some other similar conjectures.

Links

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

Example

a(10) = 1 since prime(3) + phi(7) = 5 + 6 = 11 is a Sophie prime and 2 is a primitive root modulo 11.
a(79) = 1 since prime(19) + phi(60) = 67 + 16 = 83 is a Sophie Germain prime and 2 is a primitive root modulo 83.

Mathematica

p[n_]:=PrimeQ[n]&&PrimeQ[2n+1]&&PrimitiveRoot[n]==2
f[n_, k_]:=Prime[k]+EulerPhi[n-k]
a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A001122, A005384.

Sequence in context: A129447 A125079 A329027 * A104597 A182936 A340503

Adjacent sequences: A235984 A235985 A235986 * A235988 A235989 A235990

Keyword

nonn

Author

Zhi-Wei Sun, Jan 17 2014

Status

approved