A229708 -id:A229708 - cp's OEIS frontend

A111076 Smallest positive number of maximal order mod n.

1, 1, 2, 3, 2, 5, 3, 3, 2, 3, 2, 5, 2, 3, 2, 3, 3, 5, 2, 3, 2, 7, 5, 5, 2, 7, 2, 3, 2, 7, 3, 3, 2, 3, 2, 5, 2, 3, 2, 3, 6, 5, 3, 3, 2, 5, 5, 5, 3, 3, 5, 7, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 7, 5, 5, 5, 2, 3, 2, 7, 3, 3, 2, 7, 2, 5, 3, 3, 2, 3, 3, 7, 2, 3, 11, 5, 2, 5, 5, 3, 2, 3

Offset: 1

Franklin T. Adams-Watters, Oct 10 2005

			a(6)=5 because order of 1 is 1 and 2 through 4 are not relatively prime to 6, but 5 has order 2, which is the maximum possible.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002322 (orders); same as A046145 for n with primitive roots; see also A001918 (for primes), A229708.

Table[Min[
  Select[Range[n],
   CoprimeQ[#, n] &&
     MultiplicativeOrder[#, n] == CarmichaelLambda[n] &]], {n, 1, 100}]
(* Geoffrey Critzer, Jan 04 2015 *)

a(n)=if(n==1, return(1)); if(n<5,return(n-1)); my(o=lcm(znstar(n)[2]),k=1); while(gcd(k++,n)>1 || znorder(Mod(k,n))Charles R Greathouse IV, Jul 31 2013

a(n) = A229708(n) if and only if a(n) is prime. - Jonathan Sondow, May 17 2017

A229710 Least m of maximal order mod n such that m is a sum of two squares.

Original entry on oeis.org

2, 5, 5, 5, 2, 13, 2, 5, 2, 5, 2, 5, 5, 5, 2, 13, 2, 13, 5, 5, 2, 37, 2, 5, 2, 13, 13, 5, 2, 5, 2, 5, 2, 13, 2, 13, 13, 5, 5, 13, 2, 5, 5, 5, 5, 13, 5, 37, 2, 5, 2, 5, 2, 37, 2, 13, 2, 13, 2, 5, 2, 5, 2, 5, 2, 17, 13, 5, 5, 5, 2, 13, 2, 37, 29, 13, 2, 13, 2, 5

Offset: 5

Eric M. Schmidt, Sep 27 2013

nonn

The sequence is undefined at n=4, as all the primitive roots are congruent to 3 mod 4.
Terms are not necessarily prime. For example, a(109) = 10.
a(prime(n)) = A229709(n).

			The integer 5 = 2^2 + 1^2 has order 2 mod 12, the maximum, so a(12) = 5.

Eric M. Schmidt, Table of n, a(n) for n = 5..10000
Christopher Ambrose, On the Least Primitive Root Expressible as a Sum of Two Squares, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A55, 2013.

Cf. A001481, A111076, A229708, A229709.

def A229710(n) : m = Integers(n).unit_group_exponent(); return 0 if n==1 else next(i for i in PositiveIntegers() if mod(i,n).is_unit() and mod(i,n).multiplicative_order() == m and all(p%4 != 3 or e%2==0 for (p,e) in factor(i)))

cp's OEIS Frontend

A111076 Smallest positive number of maximal order mod n.

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