A007885 Numbers n such that balanced sequences exist with n distinct elements.

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

Offset: 1

Author

Colin Mallows

Keywords

• nonn
• nice

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

Examples

			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.
		

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.

Programs

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]]]

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

Extensions

More terms and additional comments from Dean Hickerson, Sep 20 2001