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A242748 Number of ordered ways to write n = k + m with 0 < k <= m such that k is a primitive root modulo prime(k) and m is a primitive root modulo prime(m). 11
0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 3, 3, 2, 2, 3, 3, 2, 1, 2, 3, 3, 2, 3, 3, 3, 3, 1, 2, 3, 3, 3, 2, 1, 4, 2, 3, 3, 3, 3, 2, 5, 3, 4, 2, 4, 6, 6, 1, 5, 4, 6, 7, 4, 6, 4, 6, 3, 6, 3, 7, 5, 5, 6, 7, 4, 6, 8, 5, 6, 4, 6, 4, 8, 3, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Conjecture: a(n) > 0 for all n > 1.

This implies that there are infinitely many positive integers k which is a primitive root modulo prime(k).

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..8000

EXAMPLE

a(6) = 1 since 6 = 3 + 3 with 3 a primitive root modulo prime(3) = 5.

a(7) = 1 since 7 = 1 + 6 with 1 a primitive root modulo prime(1) = 2 and 6 a primitive root modulo prime(6) = 13.

a(15) = 1 since 15 = 2 + 13 with 2 a primitive root modulo prime(2) = 3 and 13 a primitive root modulo prime(13) = 41.

a(38) = 1 since 38 = 10 + 28 with 10 a primitive root modulo prime(10) = 29 and 28 a primitive root modulo prime(28) = 107.

a(53) = 1 since 53 = 3 + 50 with 3 a primitive root modulo prime(3) = 5 and 50 a primitive root modulo prime(50) = 229.

MATHEMATICA

dv[n_]:=Divisors[n]

Do[m=0; Do[Do[If[Mod[k^(Part[dv[Prime[k]-1], i]), Prime[k]]==1, Goto[aa]], {i, 1, Length[dv[Prime[k]-1]]-1}]; Do[If[Mod[(n-k)^(Part[dv[Prime[n-k]-1], j]), Prime[n-k]]==1, Goto[aa]], {j, 1, Length[dv[Prime[n-k]-1]]-1}]; m=m+1; Label[aa]; Continue, {k, 1, n/2}]; Print[n, " ", m]; Continue, {n, 1, 80}]

CROSSREFS

Cf. A000040, A000720, A242750, A242752.

Sequence in context: A170979 A025864 A070242 * A266012 A202111 A187759

Adjacent sequences:  A242745 A242746 A242747 * A242749 A242750 A242751

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 21 2014

STATUS

approved

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Last modified October 22 04:25 EDT 2019. Contains 328315 sequences. (Running on oeis4.)