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 A007885 Numbers n such that balanced sequences exist with n distinct elements. 1
 1, 2, 3, 4, 5, 7, 11, 13, 19, 23, 29, 37, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 131, 139, 149, 163, 167, 173, 179, 181, 191, 197, 199, 211, 227, 239, 263, 269, 271, 293, 311, 317, 347, 349, 359, 367, 373, 379, 383, 389, 419, 421, 443, 461, 463, 467 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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. 1, 2, 4, and primes p such that either +2 or -2 (or both) are primitive roots mod p. - Joerg Arndt, Jun 03 2012 LINKS Fred Galvin, Problem 10430, Amer. Math. Monthly, 102 (1995), 71. Fred Galvin, John Isbell and Robin J. Chapman, Problem 10430 solution, Amer. Math. Monthly, 104 (1997), 671-672. EXAMPLE 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. MATHEMATICA 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]]] PROG (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 CROSSREFS Sequence in context: A175787 A073019 A174291 * A192586 A003037 A259466 Adjacent sequences:  A007882 A007883 A007884 * A007886 A007887 A007888 KEYWORD nonn,nice AUTHOR EXTENSIONS More terms and additional comments from Dean Hickerson, Sep 20 2001 STATUS approved

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Last modified February 23 23:48 EST 2020. Contains 332195 sequences. (Running on oeis4.)