

A261773


Number of full reptend primes p < n in base n.


1



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

2,4


COMMENTS

Gives the number of primes p < n, such that the decimal expansion of 1/p has period p1, which is the greatest period possible for any integer.
Full reptend primes are also called long period primes, long primes, or maximal period primes.
Even square n have a(n) = 0, odd square n have a(n) = 1, since 2 is a full reptend prime for all odd n.
Odd n have a(n) >= 1, since 2 is a full reptend prime in all odd n whose period is 1, i.e., the maximal period (p  1).
Are 2 and 6 the only numbers other than even squares for which a(n) = 0? Are 3, 10 and 14 the only numbers other than odd squares for which a(n) = 1?  Robert Israel, Aug 31 2015


REFERENCES

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 6th ed., Oxford Univ. Press, 2008, pp. 144148.


LINKS

Robert Israel, Table of n, a(n) for n = 2..10000
OEIS Wiki, Full reptend primes.
Eric Weisstein's World of Mathematics, Cyclic Number.
Eric Weisstein's World of Mathematics, Full Reptend Prime.


EXAMPLE

a(10) = 1 since the only full reptend prime in base 10 less than 10 is 7.
a(17) = 5 since the full reptend primes {2, 3, 5, 7, 11} in base 17 are all less than 17.


MAPLE

f:= proc(n) nops(select(p > isprime(p) and numtheory:order(n, p) = p1, [$2..n1])) end proc:
map(f, [$2..100]); # Robert Israel, Aug 31 2015


MATHEMATICA

Count[Prime@ Range@ PrimePi@ #, n_ /; MultiplicativeOrder[#, n] == n  1] & /@ Range[2, 99] (* Michael De Vlieger, Aug 31 2015 *)


PROG

(PARI) a(n) = sum(k=2, n1, if (isprime(k) && (n%k), znorder(Mod(n, k))==(k1))); \\ Michel Marcus, Sep 04 2015


CROSSREFS

Cf. A001913.
Sequence in context: A163542 A061895 A129678 * A339733 A226207 A226324
Adjacent sequences: A261770 A261771 A261772 * A261774 A261775 A261776


KEYWORD

nonn,base


AUTHOR

Michael De Vlieger, Aug 31 2015


STATUS

approved



