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A078611 Radius of the shortest interval (of positive length) centered at prime(n) that has prime endpoints. 15
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 (list; graph; refs; listen; history; text; internal format)
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
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 A339743
KEYWORD
nonn,easy
AUTHOR
Joseph L. Pe, Dec 09 2002
STATUS
approved

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Last modified March 29 09:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)