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A096693
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Balance index of each prime.
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21
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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
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1,16
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COMMENTS
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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).
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LINKS
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C. H. Gribble, Table of n, a(n) for n=1,..., 10000.
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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PROG
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(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); ) ; }
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Robert G. Wilson v, Jun 26 2004
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EXTENSIONS
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Corrected and edited by Christopher Hunt Gribble, Apr 06 2010
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STATUS
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approved
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