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A218333
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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.
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0
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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
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2,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(6)-a(38) were calculated by Zak Seidov, Oct 28 2012
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STATUS
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approved
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