A078611 Radius of the shortest interval (of positive length) centered at prime(n) that has prime endpoints.

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, 24, 24, 18, 30, 12, 18, 12, 6, 36, 30, 6, 12, 18, 42, 30, 30, 42, 12, 60, 30, 48, 6, 12, 30, 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

Offset: 3

Author

Joseph L. Pe, Dec 09 2002

Keywords

• nonn
• easy

Comments

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.

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

Examples

			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.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 3..40000

Crossrefs

Cf. A047160, A082467. - Stanislav Sykora, Mar 14 2014

Programs

Mathematica

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}]

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

Formula

a(n) = A082467(A000040(n)). - Jason Kimberley, Jun 25 2012