1,1

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...

(PARI program to test the conjecture) 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,n)); if(isprime(j), break)));i

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 neigboring 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). [From Alexander R. Povolotsky (pevnev(AT)juno.com), Sep 26 2008]

(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

Cf. A000040

Cf. A144831 [From Alexander R. Povolotsky (pevnev(AT)juno.com), Sep 25 2008]

Sequence in context: A023204 A045352 A107426 this_sequence A082555 A160794 A092059

Adjacent sequences: A144253 A144254 A144255 this_sequence A144257 A144258 A144259

nonn

Alexander R. Povolotsky (pevnev(AT)juno.com), Sep 16 2008