A094901 Positive integer values of the integer Schwarzian derivatives of the primes.

0, 0, 3, 0, 3, 0, 0, 9, 0, 1, 1, 0, 0, 0, 8, 0, 1, 1, 0, 1, 0, 0, 3, 0, 0, 3, 0, 0, 14, 1, 9, 0, 32, 1, 0, 0, 0, 0, 8, 0, 32, 2, 3, 0, 0, 8, 1, 0, 0, 9, 0, 2, 0, 0, 8, 0, 1, 1, 0, 0, 12, 2, 0, 0, 5, 0, 30, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 29, 0, 32, 1, 1, 0, 0, 3, 0, 0, 0, 1, 1, 0, 3, 0, 0, 45, 0, 10, 1, 2, 0

Offset

4,3

Comments

Negative values of the integer Schwarzian derivatives of Primes are much larger in magnitude than positives values. The significance of this seems to be in its relationship to zeta zeros on the complex plane.

Links

Table of n, a(n) for n=4..104.

Formula

a(n) = Floor[Abs[IntegerSchwarzianDerivative[Prime[n]]]]

Mathematica

(* Ulam-Newton integer derivatives: *) f1[n_]=Prime[n]-Prime[n-1] f2[n_]=Prime[n]-2*Prime[n-1]+Prime[n-2] f3[n_]=Prime[n]-3*Prime[n-1]+3*Prime[n-2]-Prime[n-3] (* Integer Schwarzian derivative:*) sf[n_]=f3[n]/f1[n]-1.5*(f2[n]/f1[n])^2 af=Table[sf[n], {n, 4, 204}] a=Floor[Abs[af]]

Crossrefs

Sequence in context: A165951 A300288 A340555 * A030220 A219240 A349612

Adjacent sequences: A094898 A094899 A094900 * A094902 A094903 A094904

Keyword

nonn

Author

Roger L. Bagula, Jun 15 2004

Status

approved