A261387 Number of ways to write n = k + m with 0 < k < m < n such that prime(k) is a primitive root modulo prime(m) and also prime(m) is a primitive root modulo prime(k).

0, 0, 1, 1, 1, 1, 2, 0, 2, 1, 3, 3, 1, 1, 2, 1, 2, 7, 4, 2, 1, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 7, 3, 3, 5, 5, 5, 5, 4, 3, 6, 7, 5, 5, 5, 3, 7, 7, 5, 2, 7, 6, 4, 5, 5, 7, 10, 9, 8, 8, 4, 7, 5, 11, 14, 7, 12, 11, 9, 6

Offset

1,7

Comments

Conjecture: (i) a(n) > 0 except for n = 1, 2, 8.

(ii) Any positive rational number r not equal to 1 can be written as m/n, where m and n are positive integers such that prime(m) is a primitive root modulo prime(n) and also prime(n) is a primitive root modulo prime(m).

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

Zhi-Wei Sun, Checking part (ii) of the conjecture for r = a/b with 1 <= a < b <= 100

Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Example

a(7) = 2 since 7 = 1+6 = 3+4, prime(1) = 2 is a primitive root modulo prime(6) = 13 and 13 is a primitive root modulo 2, also prime(3) = 5 is a primitive root modulo prime(4) = 7 and 7 is a primitive root modulo 5.
a(22) = 1 since 22 = 4+18, prime(4)= 7 is a primitive root modulo prime(18) = 61 and 61 is a primitive root modulo 7.

Mathematica

f[n_]:=Prime[n]
Dv[n_]:=Divisors[n]
LL[n_]:=Length[Dv[n]]
Do[r=0; Do[Do[If[Mod[f[k]^(Part[Dv[f[n-k]-1], i])-1, f[n-k]]==0, Goto[bb]], {i, 1, LL[f[n-k]-1]-1}]; Do[If[Mod[f[n-k]^(Part[Dv[f[k]-1], i])-1, f[k]]==0, Goto[bb]], {i, 1, LL[f[k]-1]-1}];
r=r+1; Label[bb]; Continue, {k, 1, (n-1)/2}]; Print[n, " ", r]; Continue, {n, 1, 70}]

Crossrefs

Cf. A000040, A242748, A259492.

Sequence in context: A035373 A240138 A341975 * A197317 A035573 A361850

Adjacent sequences: A261384 A261385 A261386 * A261388 A261389 A261390

Keyword

nonn

Author

Zhi-Wei Sun, Aug 27 2015

Status

approved