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 A096693 Balance index of each prime. 21
 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 (list; graph; refs; listen; history; text; internal format)
 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: A204060 A085393 A128980 * A193139 A083206 A069531 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

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)