

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
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OFFSET

1,16


COMMENTS

a(n) = the number of values of k for which the nth 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) bfile 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



