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 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,4 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 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. 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) = 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 *) PROG (PARI) a(n) = sum(k=2, n-1, if (isprime(k) && (n%k), znorder(Mod(n, k))==(k-1))); \\ 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

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Last modified July 28 12:43 EDT 2021. Contains 346328 sequences. (Running on oeis4.)