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A242345 Number of primes p < prime(n) with p and 2^p - p both primitive roots modulo prime(n). 5
0, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 4, 4, 7, 1, 2, 1, 1, 1, 6, 4, 1, 4, 2, 6, 3, 7, 1, 3, 7, 4, 6, 1, 5, 6, 9, 12, 7, 5, 6, 4, 11, 2, 3, 6, 12, 6, 18, 13, 3, 14, 13, 14, 15, 4, 9, 6, 3, 13, 8, 12, 5, 12, 6, 6, 20, 8, 14, 19, 8, 5, 5, 22, 20, 6, 18, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Conjecture: a(n) > 0 for all n > 1. In other words, for any odd prime p, there is a prime q < p such that both q and 2^q - q are primitive roots modulo p.

According to page 377 in Guy's book, Erdős asked whether for any sufficiently large prime p there exists a prime q < p which is a primitive root modulo p.

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, New York, 2004.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..3500

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

EXAMPLE

a(4) = 1 since 3 is a prime smaller than prime(4) = 7, and both 3 and 2^3 - 3 = 5 are primitive roots modulo 7.

a(10) = 1 since 2 is a prime smaller than prime(10) = 29, and 2 and 2^2 - 2 are primitive roots modulo 29.

a(36) = 1 since 71 is a prime smaller than prime(36) = 151, and both 71 and 2^(71) - 71 ( == 14 (mod 151)) are primitive roots modulo 151.

MATHEMATICA

f[k_]:=2^(Prime[k])-Prime[k]

dv[n_]:=Divisors[n]

Do[m=0; Do[If[Mod[f[k], Prime[n]]==0, Goto[aa], Do[If[Mod[(Prime[k])^(Part[dv[Prime[n]-1], i]), Prime[n]]==1||Mod[f[k]^(Part[dv[Prime[n]-1], i]), Prime[n]]==1, Goto[aa]], {i, 1, Length[dv[Prime[n]-1]]-1}]]; m=m+1; Label[aa]; Continue, {k, 1, n-1}];

Print[n, " ", m]; Continue, {n, 1, 80}]

CROSSREFS

Cf. A000040, A000325, A234972, A236966, A242248, A242250, A242292.

Sequence in context: A023591 A165661 A107711 * A179067 A061893 A078530

Adjacent sequences:  A242342 A242343 A242344 * A242346 A242347 A242348

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 11 2014

STATUS

approved

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Last modified January 20 08:54 EST 2022. Contains 350471 sequences. (Running on oeis4.)