A218769 Let (p,p+2) be the n-th twin prime pair. a(n) is the least integer r > 1 for which the interval (r*p, r*(p+2)) contains no primes, or a(n)=0, if no such r exists.
0, 0, 0, 0, 4, 0, 2, 2, 2, 2, 3, 2, 5, 5, 4, 5, 4, 4, 3, 2, 2, 4, 4, 2, 2, 2, 6, 3, 3, 4, 3, 2, 3, 2, 2, 7, 3, 3, 2, 2, 2, 6, 0, 3, 2, 2, 5, 5, 23, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 5, 2
Offset: 1
Keywords
Examples
The 13th twin prime pair is {179, 181}. For r = 2 the range {358, ..., 362} contains prime 359; for r = 3, the range {537, ..., 543} contains prime 541; for r = 4, the range {716, ..., 724} contains prime 719. But for r = 5, the range {895, ..., 905} does not contain any prime. Thus a(13) = 5.
Links
- Peter J. C. Moses, Table of n, a(n) for n = 1..20000
- Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785
- J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2
Programs
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Mathematica
rmax = 100; p1[1] = 3; p1[n_] := p1[n] = (p = NextPrime[p1[n-1]]; While[ !PrimeQ[p+2], p = NextPrime[p]]; p); a[n_] := Catch[ For[r = 2, r <= rmax, r++, If[ PrimePi[r*p1[n]] == PrimePi[r*(p1[n] + 2)], Throw[r], If[r == rmax, Throw[0]]]]]; Table[ a[n] , {n, 1, 65}] (* Jean-François Alcover, Dec 13 2012 *)
Extensions
Typo in definition corrected by Jonathan Sondow, Dec 21 2012
Comments