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A145281
a(n) is the least prime such that (ceiling(sqrt(a(n)*p_n)))^2 - a(n)*p_n is a perfect square, where p_n is the n-th prime.
2
2, 3, 3, 3, 5, 5, 11, 11, 13, 17, 19, 23, 29, 29, 31, 37, 41, 41, 47, 53, 53, 59, 61, 67, 73, 79, 79, 83, 83, 89, 101, 101, 107, 109, 127, 127, 127, 131, 137, 139, 149, 149, 157, 157, 163, 163, 173, 191, 191, 191, 193, 199, 211, 211, 223, 223, 227, 227, 233, 239
OFFSET
1,1
COMMENTS
Theorem. p_n - 2*sqrt(2p_n) + 2<a(n) <= p_n. Corollary. a(n)~n*log(n). Conjecture. a(n) is the nearest prime p >= A145236(n).
Or a(n) is the least prime q_n <= p_n such that sqrt(p_n) - sqrt(q_n) < sqrt(2) [or (p_n + q_n)/2 < sqrt(p_n*q_n) + 1]. See also our comment to A145300. - Vladimir Shevelev, Oct 09 2008
The above conjecture is true. This means that a(n) is the nearest prime p > p_n - 2*floor(sqrt(2*p_n)) + 2. A considerably more important and deep question is whether p < p_n. The answer does not follow even from the Riemann conjecture about zeros of the zeta function. - Vladimir Shevelev, Oct 17 2008
PROG
(PARI) a(n) = {my(p = prime(n)); my(q = 2); while (! issquare(ceil(sqrt(q*p))^2 - q*p), q = nextprime(q+1)); q; } \\ Michel Marcus, Jul 06 2015
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Oct 06 2008, Oct 07 2008
EXTENSIONS
More terms from Michel Marcus, Jul 06 2015
STATUS
approved