

A078611


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


13



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
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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 = 75 = 2. Hence a(3) = 2. prime(5) = 11 is the center of the interval [5,17] that has prime endpoints; this interval has radius = 1711 = 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/GP) StrongGoldbachForPrimes(nmax)= {local(v, i, p, k); v=vector(nmax); for (i=1, nmax, p=prime(i); v[i] = 1; for (k=1, p2, if (isprime(pk)&&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: A174342 A111150 A166983 * A211376 A131450 A114218
Adjacent sequences: A078608 A078609 A078610 * A078612 A078613 A078614


KEYWORD

nonn,easy


AUTHOR

Joseph L. Pe, Dec 09 2002


STATUS

approved



