A161623 Greatest k for which the Andrica-like conjectural inequalities, prime(k+1)-prime(k)-(1/n)*sqrt(prime(k)) < 0, appear to fail, based on empirical evidence.

30, 429, 3644, 4612, 14357, 31545, 40933, 49414, 104071, 149689, 149689, 149689, 149689, 165326, 325852, 325852, 415069, 415069, 491237, 566214

Offset

1,1

Comments

This is a family of increasingly restrictive Andrica-like conjectures that all imply Legendre's conjecture.

Links

Table of n, a(n) for n=1..20.

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

Cf. A084976.

Sequence in context: A110612 A290363 A161740 * A210431 A296919 A010946

Adjacent sequences: A161620 A161621 A161622 * A161624 A161625 A161626

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