OEIS A261773

by Michael Thomas De Vlieger, updated 31 August 2015, St. Louis, Missouri. First published 31 August 2015.

Name	Number of full reptend primes p < n in base n.
Data	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, 4
Comments	Counts 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. Counts the number of primes p < n such that the corresponding entry in A002371 is p-1. 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.
References	G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 115.
Links	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.
Mathematica	Count[Prime@ Range@ PrimePi@ #, n_ /; MultiplicativeOrder[#, n] == n - 1] & /@ Range[2, 99] (* Michael De Vlieger, Aug 31 2015 *)
Crossrefs	A001913.
Keyword	nonn
Author	Michael De Vlieger, Aug 31 2015

This sequence simply counts the number of full reptend primes p in base n less than n itself. A full reptend prime is one that has a unit fraction 1/p which repeats a (p − 1) digit "mantissa" or "word" under expansion in base n. This is the maximum number of digits a prime can repeat to represent the fraction 1/p in base n.

The "fraction map" below charts the unit fractions with denominators in blue on the x-axis and bases n in red on the y-axis. The reptends of full reptend primes are shaded darker gray, except the light purple for p = 2 for odd bases n are also full reptends. The terminating "word" of regular numbers (OEIS A045763) appears in red for divisors of n, or orange for nondivisor regulars (OEIS A243822) of n. The former terminate in a single digit while the latter require more than one digit to terminate. The mixed recurrent "word" for semi-coprime numbers (OEIS A243823, i.e., those that have at least one prime factor p that divides n and one prime factor q that is coprime to n) appears in yellow. For these semi-coprime numbers of base n, the terminating "preamble" is written in bold red and the recurrent mantissa in italic overlined numerals. The purely recurrent reptend for numbers coprime to n appears in muted colors. Full reptends are dark gray while reptends with periods 2 < λ < (p − 1) appear in light gray. Reptends with period 1 pertaining to unit fractions of factors of (n − 1) appear in light blue, reptends with period 2 pertaining to unit fractions of factors of (n + 1) appear in light green. Light purple boxes indicate the full reptend associated with 2 for odd n; 2 is a full reptend prime in odd n but has a period of 1, which is maximal for p = 2.

This sequence in effect counts the dark gray and light purple boxes that appear to the left of the diagonal line of red "1"s. Odd n will have a(n) ≥ 1.