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

3,1

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

Links

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

Formula

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

Example

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.

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}]

Prog

(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

Crossrefs

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

Sequence in context: A111150 A166983 A361689 * A211376 A278249 A380341

Adjacent sequences: A078608 A078609 A078610 * A078612 A078613 A078614

Keyword

nonn,easy

Author

Joseph L. Pe, Dec 09 2002

Status

approved