A096693 Balance index of each prime.

0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 4, 0, 0, 5, 1, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 1, 0, 1, 0, 1, 0, 2, 0, 2, 1, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Author

Robert G. Wilson v, Jun 26 2004

Keywords

• nonn

Comments

a(n) = the number of values of k for which the n-th prime is equal to the arithmetic average of the k primes above and below it.

The average of the first n balance indexes appears to reach a global maximum of 0.588 when n = 85, (prime(85) = 439).

Examples

			a(3) = 1 because the third prime, 5, equals (3 + 7)/2.
a(16) = 3 because the sixteenth prime, 53, equals (47 + 59)/2 = (41 + 43 + 47 + 59 + 61 + 67)/6 = (31 + 37 + 41 + 43 + 47 + 59 + 61 + 67 + 71 + 73)/10.
		

Links

• C. H. Gribble, Table of n, a(n) for n=1,..., 10000.

Crossrefs

Cf. A090403, A096695, A096705, A096706, A096707, A096708, A096709, A096711.

Programs

Mathematica

f[n_] := Block[{c = 0, k = 1, p = Prime[n], s = Plus @@ Table[ Prime[i], {i, n - 1, n + 1}]}, While[k != n - 1, If[s == (2k + 1)p, c++ ]; k++; s = s + Prime[n - k] + Prime[n + k]]; c]; Table[ f[n], {n, 105}]

PARI

b-file generator: {max_n = 10^4; for (n = 1, max_n, c = 0; k = 1; p = prime(n); s = p; while (k < n, s = s + prime(n - k) + prime(n + k); if (s == (2 * k + 1) * p, c++); k++;); print(n " " c);) ;}

Extensions

Corrected and edited by Christopher Hunt Gribble, Apr 06 2010