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%I #17 Mar 24 2017 04:41:26
%S 5,53,173,211,1511,3307,3637,4457,4993,6863,11411,11731,11903,12653,
%T 15907,18223,20107,20201,20347,20731,22051,23801,26041,35911,39113,
%U 40493,46889,47303,51551,52529,60083,63559,69623,71011,75787,77081,78803,85049,91297
%N Balanced primes which are the average of two successive semiprimes.
%C Prime p which is the average of the previous prime and the following prime and is also the average of two successive semiprimes.
%H Alois P. Heinz, <a href="/A212820/b212820.txt">Table of n, a(n) for n = 1..1000</a>
%F { A212820 } = { A006562 } intersection { A103654 }.
%e 53 is in the sequence because it is the average of 47 and 59 (the two neighboring primes) and 51 and 55 (the two neighboring semiprimes).
%p with(numtheory):
%p prevsp:= proc(n) local k; for k from n-1 by -1
%p while isprime(k) or bigomega(k)<>2 do od; k end:
%p nextsp:= proc(n) local k; for k from n+1
%p while isprime(k) or bigomega(k)<>2 do od; k end:
%p a:= proc(n) option remember; local p;
%p p:= `if`(n=1, 2, a(n-1));
%p do p:= nextprime(p);
%p if p=(prevprime(p)+nextprime(p))/2 and
%p p=(prevsp(p)+nextsp(p))/2 then break fi
%p od; p
%p end:
%p seq (a(n), n=1..40); # _Alois P. Heinz_, Jun 03 2012
%t prevsp[n_] := Module[{k}, For[k = n-1, PrimeQ[k] || PrimeOmega[k] != 2, k--]; k];
%t nextsp[n_] := Module[{k}, For[k = n+1, PrimeQ[k] || PrimeOmega[k] != 2 , k++]; k];
%t a[n_] := a[n] = Module[{p}, p = If[n==1, 2, a[n-1]]; While[True, p = NextPrime[p]; If[p == (NextPrime[p, -1] + NextPrime[p])/2 && p == (prevsp[p] + nextsp[p])/2, Break[]]]; p];
%t Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Mar 24 2017, after _Alois P. Heinz_ *)
%Y Cf. A006562, A103654.
%K nonn
%O 1,1
%A _Gerasimov Sergey_, May 28 2012
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