

A237121


Number of primes p < prime(n)/2 such that P(p) is a primitive root modulo prime(n), where P(.) is the partition function given by A000041.


7



0, 0, 1, 1, 2, 2, 2, 3, 3, 5, 1, 3, 4, 1, 4, 5, 5, 5, 3, 4, 6, 6, 5, 7, 6, 8, 5, 8, 5, 8, 10, 9, 9, 9, 11, 7, 6, 9, 11, 9, 14, 5, 6, 4, 10, 4, 6, 7, 12, 9, 14, 9, 8, 11, 11, 17, 23, 11, 15, 6, 13, 22, 14, 14, 11, 19, 11, 7, 22, 13
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OFFSET

1,5


COMMENTS

Conjecture: a(n) > 0 for all n > 2. In other words, for any prime p > 3, there is a prime q < p/2 with P(q) = A000041(q) a primitive root modulo p.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..1000


EXAMPLE

a(14) = 1 since 3 is a prime smaller than prime(14)/2 = 43/2 and P(3) = 3 is a primitive root modulo prime(14) = 43.


MATHEMATICA

f[k_]:=PartitionsP[Prime[k]]
dv[n_]:=Divisors[n]
Do[m=0; Do[If[Mod[f[k], Prime[n]]==0, Goto[aa], Do[If[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, PrimePi[(Prime[n]1)/2]}]; Print[n, " ", m]; Continue, {n, 1, 70}]


CROSSREFS

Cf. A000040, A000041, A236308, A236966, A237112.
Sequence in context: A005851 A275247 A293440 * A139821 A248972 A077563
Adjacent sequences: A237118 A237119 A237120 * A237122 A237123 A237124


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Apr 22 2014


STATUS

approved



