A261773 Number of full reptend primes p < n in base n.

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

Offset: 2

Author

Michael De Vlieger, Aug 31 2015

Keywords

• nonn
• base

Comments

Gives the number of primes p < n, such that the decimal expansion of 1/p has period p-1, 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

Examples

			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.
		

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 6th ed., Oxford Univ. Press, 2008, pp. 144-148.

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.

Crossrefs

Cf. A001913.

Programs

Maple

f:= proc(n) nops(select(p -> isprime(p) and numtheory:-order(n,p) = p-1, [$2..n-1])) 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 *)

PARI

a(n) = sum(k=2, n-1, if (isprime(k) && (n%k), znorder(Mod(n, k))==(k-1))); \\ Michel Marcus, Sep 04 2015