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%I #11 Oct 21 2012 19:39:49
%S 5,17,29,37,53,71,79,89,137,149,151,157,173,179,193,211,227,229,257,
%T 263,281,349,353,359,373,383,397,409,419,421,433,439,487,491,563,577,
%U 593,607,631,643,653,659,677,701,709,733,751,757,787,823,827,877,947,953
%N Balanced primes: Primes which are both the arithmetic mean and median of a sequence of 2k+1 consecutive primes, for some k>0.
%C Union, for all k>0, of (2k+1)-balanced prime numbers, i.e., balanced prime of order k, which are primes p_n such that (2k+1)*p_n = Sum_{i=n-k..n+k} p_i, where p_i is the i-th prime.
%H Donovan Johnson, <a href="/A090403/b090403.txt">Table of n, a(n) for n = 1..10000</a>
%e 17 is in the sequence because 17 = (7 + 11 + 13 + 17 + 19 + 23 + 29)/7, (k = 3).
%e 29 is in the sequence because 29 = (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59)/15, (k = 7).
%e 37 is a member because 37 = (7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71)/17; 7 & 71 are eight primes away from 37.
%t t[n_] := (For[k=1, !(SameQ[1/(2k+1)Sum[Prime[i], {i, n-k, n+k}], Prime[n]])&& k < n-1, k++ ];k);b[n_] := If[t[n]<n-1||SameQ[1/(2n-1)Sum[Prime[i], {i, 2n-1}], Prime[n]], t[n], 0];v={};Do[If[b[n]!=0, v=Append[v, Prime[n]]], {n, 2, 168}];v
%o (PARI) is_A090403(p)={my(s=0,n); isprime(p) & for(k=1,-1+n=primepi(p),(s+=prime(n+k)+prime(n-k)-2*p)||return(1);s>p & return)} \\ _M. F. Hasler_, Oct 21 2012
%Y Cf. A096693, A006562, A082077, A082078, A082079, A096697, A096698, A096699, A096700, A096701, A096702, A096703, A096704, A096711.
%K easy,nonn
%O 1,1
%A _Farideh Firoozbakht_, Dec 07 2003
%E Definition corrected by _Franklin T. Adams-Watters_, Apr 13 2006
%E Edited by _M. F. Hasler_, Oct 21 2012
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