%I #18 Jun 06 2019 04:20:26
%S 1,2,3,4,5,7,11,13,19,23,29,37,47,53,59,61,67,71,79,83,101,103,107,
%T 131,139,149,163,167,173,179,181,191,197,199,211,227,239,263,269,271,
%U 293,311,317,347,349,359,367,373,379,383,389,419,421,443,461,463,467
%N Numbers n such that balanced sequences exist with n distinct elements.
%C A nondecreasing sequence a_1, ..., a_n is called balanced if the n-1 quantities D(a_1,...,a_k)+D(a_(k+1),...,a_n) (1<=k<=n-1) are all equal, where D(a_1,...,a_k) is the sum of the absolute deviations of the a's from their median. Up to affine equivalence, there's a unique balanced sequence of any given length.
%C n is in the sequence iff n=1, 2, or 4, or n is prime and the multiplicative group of integers mod n is generated by -1 and 2.
%C 1, 2, 4, and primes p such that either +2 or -2 (or both) are primitive roots mod p. - _Joerg Arndt_, Jun 03 2012
%H Fred Galvin, <a href="https://www.jstor.org/stable/2974863">Problem 10430</a>, Amer. Math. Monthly, 102 (1995), 71.
%H Fred Galvin, John Isbell and Robin J. Chapman, <a href="https://www.jstor.org/stable/2975067">Problem 10430 solution</a>, Amer. Math. Monthly, 104 (1997), 671-672.
%e n=5 is in the sequence, since 0,2,3,4,6 is balanced. n=6 is not because every balanced sequence of length 6 is affinely equivalent to 0,1,2,2,3,4.
%t o2[n_] := MultiplicativeOrder[2, n]; For[n=1, True, n++, If[Mod[4, n]==0||(PrimeQ[n]&&(o2[n]==n-1|| (o2[n]==(n-1)/2&&Mod[n, 4]==3))), Print[n]]]
%o (PARI) is(n)=n<6 || (isprime(n) && (znorder(Mod(2,n))==n-1 || znorder(Mod(-2,n))==n-1)) \\ _Charles R Greathouse IV_, Nov 21 2014
%K nonn,nice
%O 1,2
%A _Colin Mallows_
%E More terms and additional comments from _Dean Hickerson_, Sep 20 2001
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