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
Examples
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.
Crossrefs
Cf. A084976.
Programs
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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 *) -
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
Extensions
a(2) corrected, name edited and more terms from Michel Marcus, Aug 17 2022
Comments