OFFSET
1,1
COMMENTS
This is a family of increasingly restrictive Andrica-like conjectures that all imply Legendre's conjecture.
EXAMPLE
For n = 1, one needs k > 30 for the inequality to hold, and it is conjectured that it holds for all k > 30. In words, the first such inequality says that we expect to see a new prime prime(k+1) between prime(k) and prime(k)+sqrt(prime(k)) for k>30.
MATHEMATICA
Block[{nn = 1500000, p, q}, Array[Set[p[#], Prime[#]] &, nn + 1]; Array[Set[q[#], (p[# + 1] - p[#])^2] &, nn]; TakeWhile[Monitor[Table[nn - LengthWhile[Table[# q[k] < p[k], {k, nn, 1, -1}], # &] &[n^2], {n, 24}], {n, k}], # < nn/2 &]] (* Michael De Vlieger, Aug 17 2022 *)
PROG
(PARI) lista(nn) = my(N=10^7, vp=primes(N), va=vector(nn)); for (n=1, nn, my(v = v=vector(N-1, k, n^2*(vp[k+1]-vp[k])^2 < vp[k])); forstep(k=N-1, 1, -1, if (!v[k], va[n] = k; break)); ); va; \\ Michel Marcus, Aug 17 2022
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Daniel Tisdale, Jun 15 2009
EXTENSIONS
a(2) corrected, name edited and more terms from Michel Marcus, Aug 17 2022
STATUS
approved