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%I #20 Dec 21 2015 02:55:45
%S 2,4,6,6,6,12,6,12,12,6,12,24,6,6,12,18,6,12,6,18,24,18,30,12,6,6,30,
%T 24,24,18,30,12,18,12,6,36,30,6,12,18,42,30,30,42,12,60,30,48,6,12,30,
%U 12,6,6,12,42,6,12,54,24,24,42,36,36,18,30,36,18,6,42,30,6,30,36,30,24,18,12
%N Radius of the shortest interval (of positive length) centered at prime(n) that has prime endpoints.
%C a(1) and a(2) are undefined. Alternatively, a(n) = least k, 1 < k < n, such that prime(n) + k and prime(n) - k are both prime. I conjecture that a(n) is defined for all n > 2. Equivalently, every prime > 3 is the average of two distinct primes.
%C a(n) embodies the difference between weak and strong Goldbach conjectures, and therefore between A047160 and A082467 which differ only for prime arguments (a(n)=A082467(prime(n)), while A047160(prime(n))=0). - _Stanislav Sykora_, Mar 14 2014
%H Stanislav Sykora, <a href="/A078611/b078611.txt">Table of n, a(n) for n = 3..40000</a>
%F a(n) = A082467(A000040(n)). - _Jason Kimberley_, Jun 25 2012
%e prime(3) = 5 is the center of the interval [3,7] that has prime endpoints; this interval has radius = 7-5 = 2. Hence a(3) = 2. prime(5) = 11 is the center of the interval [5,17] that has prime endpoints; this interval has radius = 17-11 = 6. Hence a(5) = 6.
%t f[n_] := Module[{p, k}, p = Prime[n]; k = 1; While[(k < p) && (! PrimeQ[p - k] || ! PrimeQ[p + k]), k = k + 1]; k]; Table[f[i], {i, 3, 103}]
%o (PARI) StrongGoldbachForPrimes(nmax)= {local(v,i,p,k);v=vector(nmax); for (i=1,nmax,p=prime(i);v[i] = -1; for (k=1,p-2,if (isprime(p-k)&&isprime(p+k),v[i]=k;break;););); return (v);} \\ _Stanislav Sykora_, Mar 14 2014
%Y Cf. A047160, A082467. - _Stanislav Sykora_, Mar 14 2014
%K nonn,easy
%O 3,1
%A _Joseph L. Pe_, Dec 09 2002
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