The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A218333 The index of the smallest n-isolated prime p such that p/n is not between 2 and 3 and not between the smaller and greater primes of a twin prime pair, or 0 if no such p exists. 0
 5, 5, 8, 10, 2, 12, 7, 4, 37, 23, 5, 51, 3, 6, 34, 23, 5, 57, 9, 22, 49, 66, 64, 54, 5, 56, 43, 28, 46, 116, 56, 232, 92, 170, 65, 206, 181, 379, 170, 511, 190, 416, 187, 448, 89, 143, 200, 159, 434, 670, 145, 1081, 213, 1011, 680, 77 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS For n>=2, a prime p is called n-isolated (cf. A166251 and the Shevelev link, Section 10) if there is no other prime in the interval (n*prevprime(p/n), n*nextprime(p/n)). In particular, if a(n)=1, then the smallest n-isolated prime divided by n is not between 2 and 3 and not between the smaller and greater primes of a twin prime pair. Suppose that for every n there exist infinitely many n-isolated primes. Then if there exists n_0 such that a(n_0)=0, there are infinitely many twin primes. On the other hand, one can prove that the smallest n-isolated prime divided by n tends to infinity as n goes to infinity. Therefore, if there is not an N such that, for all n >= N, a(n)=1, then we also conclude that there are infinitely many twin primes. Conjecture: all a(n) >= 2. LINKS Table of n, a(n) for n=2..57. V. Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.4 EXAMPLE Let n=2. The 2-isolated primes are in A166251. The first 2-isolated prime is 5, and 5/2 is between 2 and 3. The second 2-isolated prime is 7, and 7/2 is between 3 and 5. The third 2-isolated prime is 23, and 23/2 is between 11 and 13. The fourth 2-isolated prime is 37, and 37/2 is between 17 and 19. The fifth 2-isolated prime is 79, and 79/2 is between 37 and 41. Since (37,41) is not (2,3) and is not a twin prime pair, a(2)=5 PROG (PARI) isoki(p, n) = (p==nextprime(n*precprime(p\n))) && (p==precprime(n*nextprime(p/n))); \\ A166251 nextp(p, n) = while(! isoki(p, n), p = nextprime(p+1)); p; isokp(p, n) = {my(diff = nextprime(p/n) - precprime(p/n)); if ((diff == 1) || (diff == 2), return (0)); return (1); } a(n) = {my(p = nextp(2, n), nb = 1); while (! isokp(p, n), p = nextp(nextprime(p+1), n); nb++; ); nb; } \\ Michel Marcus, Dec 16 2018 CROSSREFS Cf. A166251, A217561, A217566. Sequence in context: A358257 A003861 A107623 * A212533 A081287 A303715 Adjacent sequences: A218330 A218331 A218332 * A218334 A218335 A218336 KEYWORD nonn AUTHOR Vladimir Shevelev, Oct 26 2012 EXTENSIONS a(6)-a(38) were calculated by Zak Seidov, Oct 28 2012 More terms from Michel Marcus, Dec 16 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 26 08:27 EDT 2024. Contains 372813 sequences. (Running on oeis4.)