

A218333


The index of the smallest nisolated 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
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OFFSET

2,1


COMMENTS

For n>=2, a prime p is called nisolated (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 nisolated 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 nisolated 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 nisolated 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 2isolated primes are in A166251.
The first 2isolated prime is 5, and 5/2 is between 2 and 3.
The second 2isolated prime is 7, and 7/2 is between 3 and 5.
The third 2isolated prime is 23, and 23/2 is between 11 and 13.
The fourth 2isolated prime is 37, and 37/2 is between 17 and 19.
The fifth 2isolated 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: A141538 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



