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A144256 a(n) = smallest prime in the range [n^K, (n+1)^K], where K = log(127)/log(16) = 1.74717117169304146332... 4
2, 5, 7, 13, 17, 23, 31, 41, 47, 59, 67, 79, 89, 101, 127, 127, 149, 157, 173, 191, 211, 223, 241, 263, 277, 307, 317, 347, 359, 383, 409, 431, 457, 479, 499, 541, 557, 577, 607, 631, 659, 691, 719, 751, 787, 809, 839, 877, 907, 937, 967, 997, 1031, 1069 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

T. D. Noe submitted to primepuzzles.net the following conjecture #60, which is stronger than the Legendre's conjecture: For n>0 and k>=K, there is always a prime between n^k and (n+1)^k, where K = log(127)/log(16) = 1.74717117169304146332...

One could see that calculated terms for n=15 and n=16 yield the same value: 127, which make this conjecture (as originally defined) to be questionable. If this conjecture is modified to k>K, then there will be a distinct prime between 15^k and 16^k. It appears that the relatively large prime gap between 113 and 127 is the largest gap to overcome. Another way to correct/clarify the conjecture is to mention that both boundaries of the interval are included and that the same prime value may appear in two neighboring intervals. Of course the last version of the modified definition makes this conjecture to be different from the original Legendere conjecture (rather than to be an improvement of the original Legendere conjecture). [Alexander R. Povolotsky, Sep 26 2008]

The constant log(127)/log(16) is A194361. - John W. Nicholson, Dec 13 2013

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

Carlos Rivera, Conjecture 60: Generalization of Legendre's Conjecture

MATHEMATICA

k = Log[127]/Log[16]; Table[Select[Range[Ceiling[n^k], Floor[(n + 1)^k]], PrimeQ, 1][[1]], {n, 100}] (* T. D. Noe, Jan 30 2012 *)

PROG

(PARI) i=[]; for(n=1, 500, for(j=ceil(n^(log(127)/log(16))), ceil((n+1)^(log(127)/log(16))), if(isprime(j), i=concat(i, j)); if(isprime(j), break))); i

CROSSREFS

Cf. A000040, A144831, A143935.

Sequence in context: A278494 A107426 A222532 * A082555 A160794 A092059

Adjacent sequences:  A144253 A144254 A144255 * A144257 A144258 A144259

KEYWORD

nonn

AUTHOR

Alexander R. Povolotsky, Sep 16 2008

EXTENSIONS

Extended by T. D. Noe, Jan 30 2012

STATUS

approved

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Last modified October 18 21:36 EDT 2018. Contains 316327 sequences. (Running on oeis4.)