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

1,16

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).

Links

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

Example

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.

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}]

Prog

(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); ) ; }

Crossrefs

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

Sequence in context: A085393 A128980 A143276 * A193139 A378440 A378439

Adjacent sequences: A096690 A096691 A096692 * A096694 A096695 A096696

Keyword

nonn

Author

Robert G. Wilson v, Jun 26 2004

Extensions

Corrected and edited by Christopher Hunt Gribble, Apr 06 2010

Status

approved