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%I #20 Nov 07 2014 04:07:12
%S 8,10,12,16,18,22,24,30,34,36,42,46,48,54,60,64,66,72,76,78,84,90,102,
%T 106,108,112,114,120,126,132,138,142,144,150,154,156,162,168,174,180,
%U 184,186,192,196,198,202,204,210,216,222,228,232,234,240,244,246,252
%N Sums of two numbers that are the average of a twin prime pair (A014574).
%C The addends need not be distinct, see example.
%C Probably a(n) ~ 6n and all multiples of 6 greater than 4206 are present. - _Charles R Greathouse IV_, Nov 06 2014
%H Charles R Greathouse IV, <a href="/A249628/b249628.txt">Table of n, a(n) for n = 1..10000</a>
%e 8 is in the sequence since 8 = 4+4 and 4 is the average of the twin primes 3 and 5.
%e 10 is in the sequence since 10 = 4+6 and both 4 and 6 are the average of the twin primes 3,5 and 5,7 respectively.
%e 12 is in the sequence since 12 = 6+6 and 6 is the average of the twin primes 5 and 7.
%e 14 is not in the sequence since it is not the sum of two numbers that are the average of twin primes.
%o (PARI) v=[];for(n=1,10^3,if(isprime(n-1)&&isprime(n+1),v=concat(v,n)))
%o b=[];for(n=1,1000,for(k=1,n,if(vecsearch(v,k)&&vecsearch(v,n-k),b=concat(b,n))));vecsort(b,,8) \\ _Derek Orr_, Nov 02 2014
%o (PARI) list(lim)=my(v=List(),p,r); p=3; forprime(q=5,lim-3, if(q-p>2, p=q; next); r=3; forprime(s=5,min(lim-q+2,q), if(s-r==2, listput(v, p+s)); r=s); p=q); Set(v) \\ _Charles R Greathouse IV_, Nov 06 2014
%o (PARI) list(lim)=my(v=List(),P=primes([3,lim\1-3]),Q=v,t); for(i=2,#P, if(P[i]-P[i-1]==2, listput(Q,P[i]-1))); P=0; forstep(n=8,lim,2, for(i=1,#Q, if(setsearch(Q,t=n-Q[i]), listput(v,n); break); if(t<Q[i], break))); Vec(v) \\ _Charles R Greathouse IV_, Nov 06 2014
%o (PARI) is(n)=my(p=3);forprime(q=5,n\2+1,if(isprime(n-p)&&isprime(n-p-2),return(1));p=q);0 \\ _Charles R Greathouse IV_, Nov 06 2014
%Y Cf. A014574 (average of twin prime pairs).
%K nonn
%O 1,1
%A _Wesley Ivan Hurt_, Nov 02 2014
%E Missing a(19) added by _Charles R Greathouse IV_, Nov 06 2014
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